Chapter 9, Problem 2P

A number of matrices are defined as

$\begin{array}{l}\left[A\right]=\left[\begin{array}{c}4\\ 1\\ 5\end{array}\begin{array}{c}7\\ 2\\ 6\end{array}\right]\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]\\ \left[C\right]=\left\{\begin{array}{c}3\\ 6\\ 1\end{array}\right\}\left[D\right]=\begin{array}{cc}9& 4\\ 2& -1\end{array}\begin{array}{cc}3& -6\\ 7& 5\end{array}\end{array}$

$\begin{array}{l}\left[E\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]\\ \left[F\right]=\left[\begin{array}{c}3\\ 1\end{array}\begin{array}{c}0\\ 7\end{array}\begin{array}{c}1\\ 3\end{array}\right]\left[G\right]=\left[\begin{array}{ccc}7& 6& 4\end{array}\right]\end{array}$

Answer the following questions regarding these matrices:

(a) What are the dimensions of the matrices?

(b) Identify the square, column, and row matrices.

(c) What are the values of the elements: $a_{12},b_{23},d_{32},e_{22},f_{12},g_{12}$?

(d) Perform the following operations:

$\begin{array}{l}\left(1\right)\left[E\right]+\left[B\right]\\ \left(2\right)\left[A\right]\times \left(F\right)\\ \left(3\right)\left[B\right]-\left[E\right]\\ \left(4\right)7\times \left[B\right]\end{array}$ $\begin{array}{l}\left(5\right)\left[E\right]\times \left[B\right]\\ \left(6\right){\left\{C\right\}}^T\\ \left(7\right)\left[B\right]\times \left[A\right]\\ \left(8\right){\left[D\right]}^T\end{array}$

$\begin{array}{l}\left(9\right)\left[A\right]\times \left\{C\right\}\left(11\right){\left[E\right]}^T\left[E\right]\\ \left(10\right)\left[l\right]\times \left[B\right]\left(12\right){\left\{C\right\}}^T\left\{C\right\}\end{array}$

To determine

To calculate: The dimensions of the matrices,

$\left[A\right]=\left[\begin{array}{ll}4\hfill & 7\hfill \\ 1\hfill & 2\hfill \\ 5\hfill & 6\hfill \end{array}\right]$, $\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$, $\left[C\right]=\left\{\begin{array}{l}3\hfill \\ 6\hfill \\ 1\hfill \end{array}\right\}$, $\left[D\right]=\left[\begin{array}{llll}9\hfill & 4\hfill & 3\hfill & -6\hfill \\ 2\hfill & -1\hfill & 7\hfill & 5\hfill \end{array}\right]$, $\left[\text{E}\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]$, $\left[F\right]=\left[\begin{array}{lll}3\hfill & 0\hfill & 1\hfill \\ 1\hfill & 7\hfill & 3\hfill \end{array}\right]$, $\left[G\right]=\left[\begin{array}{lll}7\hfill & 6\hfill & 4\hfill \end{array}\right]$.

Answer to Problem 2P

Solution:

Dimension of the matrix $\left[A\right]$ is $3\times 2$, dimension of the matrix $\left[B\right]$ is $3\times 3$, dimension of the matrix $\left[C\right]$ is $3\times 1$, dimension of the matrix $\left[D\right]$ is $2\times 4$, dimension of the matrix $\left[E\right]$ is $3\times 3$, dimension of the matrix $\left[F\right]$ is $2\times 3$ and dimension of the matrix $\left[G\right]$ is $1\times 3$.

Explanation of Solution

Given:

The matrices,

$\left[A\right]=\left[\begin{array}{ll}4\hfill & 7\hfill \\ 1\hfill & 2\hfill \\ 5\hfill & 6\hfill \end{array}\right]$, $\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$, $\left[C\right]=\left\{\begin{array}{l}3\hfill \\ 6\hfill \\ 1\hfill \end{array}\right\}$, $\left[D\right]=\left[\begin{array}{llll}9\hfill & 4\hfill & 3\hfill & -6\hfill \\ 2\hfill & -1\hfill & 7\hfill & 5\hfill \end{array}\right]$, $\left[\text{E}\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]$, $\left[F\right]=\left[\begin{array}{lll}3\hfill & 0\hfill & 1\hfill \\ 1\hfill & 7\hfill & 3\hfill \end{array}\right]$, $\left[G\right]=\left[\begin{array}{lll}7\hfill & 6\hfill & 4\hfill \end{array}\right]$.

Formula used:

The representation of dimension of any matrix can be written as $\left(\text{number of rows }\times \text{ number of columns}\right)$.

Calculation:

Consider the matrix A,

$\left[A\right]=\left[\begin{array}{ll}4\hfill & 7\hfill \\ 1\hfill & 2\hfill \\ 5\hfill & 6\hfill \end{array}\right]$

Matrix A has $3$ rows and $2$ columns.

Therefore, the dimension of the matrix A is $3\times 2$.

Consider the matrix B,

$\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$

Matrix B has $3$ rows and $3$ columns.

Therefore, the dimension of the matrix B is $3\times 3$.

Consider the matrix C,

$\left[C\right]=\left[\begin{array}{l}3\hfill \\ 6\hfill \\ 1\hfill \end{array}\right]$

Matrix C has $3$ rows and $1$ column.

Therefore, the dimension of the matrix C is $3\times 1$.

Consider the matrix D,

$\left[D\right]=\left[\begin{array}{llll}9\hfill & 4\hfill & 3\hfill & -6\hfill \\ 2\hfill & -1\hfill & 7\hfill & 5\hfill \end{array}\right]$

Matrix D has $2$ rows and $4$ columns.

Therefore, the dimension of the matrix D is $2\times 4$.

Consider the matrix E,

$\left[\text{E}\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]$

Matrix E has $3$ rows and $3$ columns.

Therefore, the dimension of the matrix E is $3\times 3$.

Consider the matrix F,

$\left[F\right]=\left[\begin{array}{lll}3\hfill & 0\hfill & 1\hfill \\ 1\hfill & 7\hfill & 3\hfill \end{array}\right]$

The matrix F has $2$ rows and $3$ columns.

Therefore, the dimension of the matrix F is $2\times 3$.

Consider the matrix G,

$\left[G\right]=\left[\begin{array}{lll}7\hfill & 6\hfill & 4\hfill \end{array}\right]$

Matrix G has $1$ row and $3$ columns.

Therefore, the dimension of the matrix G is $1\times 3$.

To determine

The square matrix, column matrix and row matrix from the following matrices,

$\left[A\right]=\left[\begin{array}{ll}4\hfill & 7\hfill \\ 1\hfill & 2\hfill \\ 5\hfill & 6\hfill \end{array}\right]$, $\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$, $\left[C\right]=\left\{\begin{array}{l}3\hfill \\ 6\hfill \\ 1\hfill \end{array}\right\}$, $\left[D\right]=\left[\begin{array}{llll}9\hfill & 4\hfill & 3\hfill & -6\hfill \\ 2\hfill & -1\hfill & 7\hfill & 5\hfill \end{array}\right]$, $\left[\text{E}\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]$, $\left[F\right]=\left[\begin{array}{lll}3\hfill & 0\hfill & 1\hfill \\ 1\hfill & 7\hfill & 3\hfill \end{array}\right]$, $\left[G\right]=\left[\begin{array}{lll}7\hfill & 6\hfill & 4\hfill \end{array}\right]$.

Answer to Problem 2P

Solution:

Matrices B and E are square matrices, matrix C is a column matrix and matrix G is a row matrix.

Explanation of Solution

Given:

The matrices,

$\left[A\right]=\left[\begin{array}{ll}4\hfill & 7\hfill \\ 1\hfill & 2\hfill \\ 5\hfill & 6\hfill \end{array}\right]$, $\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$, $\left[C\right]=\left\{\begin{array}{l}3\hfill \\ 6\hfill \\ 1\hfill \end{array}\right\}$, $\left[D\right]=\left[\begin{array}{llll}9\hfill & 4\hfill & 3\hfill & -6\hfill \\ 2\hfill & -1\hfill & 7\hfill & 5\hfill \end{array}\right]$, $\left[\text{E}\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]$, $\left[F\right]=\left[\begin{array}{lll}3\hfill & 0\hfill & 1\hfill \\ 1\hfill & 7\hfill & 3\hfill \end{array}\right]$, $\left[G\right]=\left[\begin{array}{lll}7\hfill & 6\hfill & 4\hfill \end{array}\right]$.

Square matrix: A matrix M is said to be square matrix, if number of rows equals to number of columns.

Column Matrix: A matrix M is said to be column matrix, if number of rows can be any natural number but number of columns should be $1$.

Row matrix: A matrix M is said to be row matrix, if number of rows should be $1$ but number of columns can be any natural number.

Since, the dimension of the matrix B is $3\times 3$.

So, number of rows equal to number of columns.

Hence, $\left[B\right]$ is a square matrix.

Since, the dimension of the matrix C is $3\times 1$.

So, number of rows is $3$ and number of columns is $1$.

Hence, $\left[C\right]$ is a column matrix.

Since, the dimension of the matrix E is $3\times 3$.

So, number of rows equal to number of columns.

Hence, $\left[E\right]$ is a square matrix.

Since, the dimension of the matrix G is $1\times 3$.

So, number of rows is $1$ and number of columns is $3$.

Hence, $\left[G\right]$ is a row matrix.

Therefore, matrices B and E are square matrices, matrix C is a column matrix and matrix G is a row matrix.

To determine

To calculate: The value of the elements $a_{12},b_{23},d_{32},e_{22},f_{12}{\text{ and g}}_{12}$ from the following matrices,

$\left[A\right]=\left[\begin{array}{ll}4\hfill & 7\hfill \\ 1\hfill & 2\hfill \\ 5\hfill & 6\hfill \end{array}\right]$, $\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$, $\left[C\right]=\left\{\begin{array}{l}3\hfill \\ 6\hfill \\ 1\hfill \end{array}\right\}$, $\left[D\right]=\left[\begin{array}{llll}9\hfill & 4\hfill & 3\hfill & -6\hfill \\ 2\hfill & -1\hfill & 7\hfill & 5\hfill \end{array}\right]$, $\left[\text{E}\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]$, $\left[F\right]=\left[\begin{array}{lll}3\hfill & 0\hfill & 1\hfill \\ 1\hfill & 7\hfill & 3\hfill \end{array}\right]$, $\left[G\right]=\left[\begin{array}{lll}7\hfill & 6\hfill & 4\hfill \end{array}\right]$.

Answer to Problem 2P

Solution:

The value of the elements $a_{12}=7$, $b_{23}=7$, $d_{32}=\text{ does not exist}$, $e_{22}=2$, $f_{12}=0$ and $g_{12}=6$.

Explanation of Solution

Given:

The matrices,

$\left[A\right]=\left[\begin{array}{ll}4\hfill & 7\hfill \\ 1\hfill & 2\hfill \\ 5\hfill & 6\hfill \end{array}\right]$, $\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$, $\left[C\right]=\left\{\begin{array}{l}3\hfill \\ 6\hfill \\ 1\hfill \end{array}\right\}$, $\left[D\right]=\left[\begin{array}{llll}9\hfill & 4\hfill & 3\hfill & -6\hfill \\ 2\hfill & -1\hfill & 7\hfill & 5\hfill \end{array}\right]$, $\left[\text{E}\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]$, $\left[F\right]=\left[\begin{array}{lll}3\hfill & 0\hfill & 1\hfill \\ 1\hfill & 7\hfill & 3\hfill \end{array}\right]$, $\left[G\right]=\left[\begin{array}{lll}7\hfill & 6\hfill & 4\hfill \end{array}\right]$.

Formula used:

The value of the element $a_{ij}$ is the ${\left(i,j\right)}^{th}$ value of the matrix, where $\left(i,j\right)$ denote the $i^{th}$ row and $j^{th}$ column of the matrix.

Calculation:

Consider the matrix A,

$\left[A\right]=\left[\begin{array}{ll}4\hfill & 7\hfill \\ 1\hfill & 2\hfill \\ 5\hfill & 6\hfill \end{array}\right]$

Then the value of $a_{12}$ is the ${\left(1,2\right)}^{th}$ value of the matrix A, that is $7$.

Therefore, $a_{12}=7$.

Consider the matrix B,

$\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$

Then the value of $b_{23}$ is the ${\left(2,3\right)}^{th}$ value of the matrix B, that is $7$.

Therefore, $b_{23}=7$.

Consider the matrix D,

$\left[D\right]=\left[\begin{array}{llll}9\hfill & 4\hfill & 3\hfill & -6\hfill \\ 2\hfill & -1\hfill & 7\hfill & 5\hfill \end{array}\right]$

Then the value of $d_{32}$ is the ${\left(3,2\right)}^{th}$ value of the matrix D.

Since, matrix D does not have $3^{rd}$ row. Hence, $d_{32}$ does not exist.

Therefore, $d_{32}=\text{ does not exist}$.

Consider the matrix E,

$\left[\text{E}\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]$

Then the value of $e_{22}$ is the ${\left(2,2\right)}^{th}$ value of the matrix E, that is $2$.

Therefore, $e_{22}=2$.

Consider the matrix F,

$\left[F\right]=\left[\begin{array}{lll}3\hfill & 0\hfill & 1\hfill \\ 1\hfill & 7\hfill & 3\hfill \end{array}\right]$

Then the value of $f_{12}$ is the ${\left(1,2\right)}^{th}$ value of the matrix F, that is $0$.

Therefore, $f_{12}=0$.

Consider the matrix G,

$\left[G\right]=\left[\begin{array}{lll}7\hfill & 6\hfill & 4\hfill \end{array}\right]$

Then the value of $g_{12}$ is the ${\left(1,2\right)}^{th}$ value of the matrix G, that is $6$.

Therefore, $g_{12}=6$.

To determine

To calculate: The following operations,

(1) $\left[E\right]+\left[B\right]$,

(2) $\left[A\right]\times \left[F\right]$,

(3) $\left[B\right]-\left[E\right]$,

(4) $7\times \left[B\right]$,

(5) $\left[E\right]\times \left[B\right]$,

(6) ${\left\{C\right\}}^T$,

(7) $\left[B\right]\times \left[A\right]$,

(8) ${\left[D\right]}^T$,

(9) $\left[A\right]\times \left\{C\right\}$,

(10) $\left[I\right]\times \left[B\right]$,

(11) ${\left[E\right]}^T\times \left[E\right]$ and

(12) ${\left\{C\right\}}^T\times \left\{C\right\}$.

Where,

$\left[A\right]=\left[\begin{array}{ll}4\hfill & 7\hfill \\ 1\hfill & 2\hfill \\ 5\hfill & 6\hfill \end{array}\right]$, $\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$, $\left[C\right]=\left\{\begin{array}{l}3\hfill \\ 6\hfill \\ 1\hfill \end{array}\right\}$, $\left[D\right]=\left[\begin{array}{llll}9\hfill & 4\hfill & 3\hfill & -6\hfill \\ 2\hfill & -1\hfill & 7\hfill & 5\hfill \end{array}\right]$, $\left[\text{E}\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]$, $\left[F\right]=\left[\begin{array}{lll}3\hfill & 0\hfill & 1\hfill \\ 1\hfill & 7\hfill & 3\hfill \end{array}\right]$, $\left[G\right]=\left[\begin{array}{lll}7\hfill & 6\hfill & 4\hfill \end{array}\right]$.

Answer to Problem 2P

Solution:

(1) $\left[E\right]+\left[B\right]=\left[\begin{array}{ccc}5& 8& 15\\ 8& 4& 10\\ 6& 0& 10\end{array}\right]$, (2) $\left[A\right]\times \left[F\right]=\left[\begin{array}{ccc}19& 49& 25\\ 5& 14& 7\\ 21& 42& 23\end{array}\right]$, (3) $\left[B\right]-\left[E\right]=\left[\begin{array}{ccc}3& -2& -1\\ -6& 0& 4\\ -2& 0& -2\end{array}\right]$,

(4) $7\times \left[B\right]=\left[\begin{array}{ccc}28& 21& 49\\ 7& 14& 49\\ 14& 0& 28\end{array}\right]$, (5) $\left[E\right]\times \left[B\right]=\left[\begin{array}{ccc}25& 13& 74\\ 36& 25& 75\\ 28& 12& 52\end{array}\right]$, (6) ${\left\{C\right\}}^T=\left\{\begin{array}{ccc}3& 6& 1\end{array}\right\}$,

(7) $\left[B\right]\times \left[A\right]=\left[\begin{array}{ll}54\hfill & 76\hfill \\ 41\hfill & 53\hfill \\ 28\hfill & 38\hfill \end{array}\right]$, (8) ${\left[D\right]}^T=\left[\begin{array}{ll}9\hfill & 2\hfill \\ 4\hfill & -1\hfill \\ 3\hfill & 7\hfill \\ -6\hfill & 5\hfill \end{array}\right]$, (9) $\left[A\right]\times \left\{C\right\}=\text{ not possible}$,

(10) $\left[I\right]\times \left[B\right]=\left[B\right]$, (11) ${\left[\text{E}\right]}^T\times \left[\text{E}\right]=\left[\begin{array}{ccc}66& 19& 53\\ 19& 29& 46\\ 53& 46& 109\end{array}\right]$, (12) ${\left\{C\right\}}^T\times \left\{C\right\}=\left[46\right]$.

Explanation of Solution

Given:

The matrices,

$\left[A\right]=\left[\begin{array}{ll}4\hfill & 7\hfill \\ 1\hfill & 2\hfill \\ 5\hfill & 6\hfill \end{array}\right]$, $\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$, $\left[C\right]=\left\{\begin{array}{l}3\hfill \\ 6\hfill \\ 1\hfill \end{array}\right\}$, $\left[D\right]=\left[\begin{array}{llll}9\hfill & 4\hfill & 3\hfill & -6\hfill \\ 2\hfill & -1\hfill & 7\hfill & 5\hfill \end{array}\right]$, $\left[\text{E}\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]$, $\left[F\right]=\left[\begin{array}{lll}3\hfill & 0\hfill & 1\hfill \\ 1\hfill & 7\hfill & 3\hfill \end{array}\right]$, $\left[G\right]=\left[\begin{array}{lll}7\hfill & 6\hfill & 4\hfill \end{array}\right]$.

Formula used:

If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and $B=\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]$ be any two matrices.

Addition of A and B is denoted by $A+B$ and equal to $\left[\begin{array}{cc}a+e& b+f\\ c+g& d+h\end{array}\right]$.

Subtraction of A and B is denoted by $A-B$ and equal to $\left[\begin{array}{cc}a-e& b-f\\ c-g& d-h\end{array}\right]$.

Multiply any scalar quantity $5$ in the matrix A is denoted by $5A$ and equal to $\left[\begin{array}{cc}5a& 5b\\ 5c& 5d\end{array}\right]$.

Transpose of the matrix A is denoted by $A^T$ and equal to $\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]$.

Multiplication of the matrices A and B is denoted by $\left[A\times B\right]$ and equal to $\left[\begin{array}{cc}a.e+b.g& a.f+b.h\\ c.e+d.g& c.f+d.h\end{array}\right]$.

Multiplication of two matrices is possible if interior dimensions are equal.

If matrix A has dimension $m\times n$ and matrix B has dimension $p\times q$. Then $\left[A\right]\times \left[B\right]$ is possible only if $n=p$. Interior dimensions of $\left[A\right]\times \left[B\right]$ is n and p. Exterior dimensions of $\left[A\right]\times \left[B\right]$ is m and q.

Calculation:

(1) Consider the matrices E and B,

$\left[\text{E}\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]$ and $\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$.

Now, the addition of two matrices is,

$\begin{array}{c}\left[\text{E}\right]+\left[B\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]+\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]\\ =\left[\begin{array}{ccc}1+4& 5+3& 8+7\\ 7+1& 2+2& 3+7\\ 4+2& 0+0& 6+4\end{array}\right]\\ =\left[\begin{array}{ccc}5& 8& 15\\ 8& 4& 10\\ 6& 0& 10\end{array}\right]\end{array}$

Therefore, $\left[\text{E}\right]+\left[B\right]=\left[\begin{array}{ccc}5& 8& 15\\ 8& 4& 10\\ 6& 0& 10\end{array}\right]$.

(2) Consider the matrices A and F,

$\left[A\right]=\left[\begin{array}{ll}4\hfill & 7\hfill \\ 1\hfill & 2\hfill \\ 5\hfill & 6\hfill \end{array}\right]$ and $\left[F\right]=\left[\begin{array}{lll}3\hfill & 0\hfill & 1\hfill \\ 1\hfill & 7\hfill & 3\hfill \end{array}\right]$.

Now, the multiplication of two matrices is,

$\begin{array}{c}\left[A\right]\times \left[F\right]=\left[\begin{array}{ll}4\hfill & 7\hfill \\ 1\hfill & 2\hfill \\ 5\hfill & 6\hfill \end{array}\right]\times \left[\begin{array}{lll}3\hfill & 0\hfill & 1\hfill \\ 1\hfill & 7\hfill & 3\hfill \end{array}\right]\\ =\left[\begin{array}{ccc}4\times 3+7\times 1& 4\times 0+7\times 7& 4\times 1+7\times 3\\ 1\times 3+2\times 1& 1\times 0+2\times 7& 1\times 1+2\times 3\\ 5\times 3+6\times 1& 5\times 0+6\times 7& 5\times 1+6\times 3\end{array}\right]\\ =\left[\begin{array}{ccc}19& 49& 25\\ 5& 14& 7\\ 21& 42& 23\end{array}\right]\end{array}$

Therefore, $\left[A\right]\times \left[F\right]=\left[\begin{array}{ccc}19& 49& 25\\ 5& 14& 7\\ 21& 42& 23\end{array}\right]$.

(3) Consider the matrices B and E,

$\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$ and $\left[\text{E}\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]$.

Now, the subtraction of two matrices is,

$\begin{array}{c}\left[B\right]-\left[E\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]-\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]\\ =\left[\begin{array}{ccc}4-1& 3-5& 7-8\\ 1-7& 2-2& 7-3\\ 2-4& 0-0& 4-6\end{array}\right]\\ =\left[\begin{array}{ccc}3& -2& -1\\ -6& 0& 4\\ -2& 0& -2\end{array}\right]\end{array}$

Therefore, $\left[B\right]-\left[E\right]=\left[\begin{array}{ccc}3& -2& -1\\ -6& 0& 4\\ -2& 0& -2\end{array}\right]$.

(4) Consider a matrix B, and a scalar $7$,

$\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$.

The multiplication of a matrix B with a scalar $7$ is,

$\begin{array}{c}7\times \left[B\right]=7\times \left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]\\ =\left[\begin{array}{ccc}7\times 4& 7\times 3& 7\times 7\\ 7\times 1& 7\times 2& 7\times 7\\ 7\times 2& 7\times 0& 7\times 4\end{array}\right]\\ =\left[\begin{array}{ccc}28& 21& 49\\ 7& 14& 49\\ 14& 0& 28\end{array}\right]\end{array}$

Therefore, $7\times \left[B\right]=\left[\begin{array}{ccc}28& 21& 49\\ 7& 14& 49\\ 14& 0& 28\end{array}\right]$.

(5) Consider the matrices E and B,

$\left[\text{E}\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]$ and $\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$.

Now the multiplication of two matrices E and B is,

$\begin{array}{c}\left[E\right]\times \left[B\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]\times \left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]\\ =\left[\begin{array}{ccc}1\times 4+5\times 1+8\times 2& 1\times 3+5\times 2+8\times 0& 1\times 7+5\times 7+8\times 4\\ 7\times 4+2\times 1+3\times 2& 7\times 3+2\times 2+3\times 0& 7\times 7+2\times 7+3\times 4\\ 4\times 4+0\times 1+6\times 2& 4\times 3+0\times 2+6\times 0& 4\times 7+0\times 7+6\times 4\end{array}\right]\\ =\left[\begin{array}{ccc}25& 13& 74\\ 36& 25& 75\\ 22& 12& 52\end{array}\right]\end{array}$

Therefore, $\left[E\right]\times \left[B\right]=\left[\begin{array}{ccc}25& 13& 74\\ 36& 25& 75\\ 22& 12& 52\end{array}\right]$.

(6) Consider the matrix C,

$\left[C\right]=\left\{\begin{array}{l}3\hfill \\ 6\hfill \\ 1\hfill \end{array}\right\}$

Now the transpose of the matrix C is,

${\left[C\right]}^T=\left\{\begin{array}{ccc}3& 6& 1\end{array}\right\}$

Therefore, ${\left[C\right]}^T=\left\{\begin{array}{ccc}3& 6& 1\end{array}\right\}$.

(7) Consider the matrices B and A,

$\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$ and $\left[A\right]=\left[\begin{array}{ll}4\hfill & 7\hfill \\ 1\hfill & 2\hfill \\ 5\hfill & 6\hfill \end{array}\right]$

Now the multiplication of the two matrices B and A is,

$\begin{array}{c}\left[B\right]\times \left[A\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]\times \left[\begin{array}{ll}4\hfill & 7\hfill \\ 1\hfill & 2\hfill \\ 5\hfill & 6\hfill \end{array}\right]\\ =\left[\begin{array}{cc}4\times 4+3\times 1+7\times 5& 4\times 7+3\times 2+7\times 6\\ 1\times 4+2\times 1+7\times 5& 1\times 7+2\times 2+7\times 6\\ 2\times 4+0\times 1+4\times 5& 2\times 7+0\times 2+4\times 6\end{array}\right]\\ =\left[\begin{array}{cc}54& 76\\ 41& 53\\ 28& 38\end{array}\right]\end{array}$

Therefore, $\left[B\right]\times \left[A\right]=\left[\begin{array}{cc}54& 76\\ 41& 53\\ 28& 38\end{array}\right]$

(8) Consider the matrix D,

$\left[D\right]=\left[\begin{array}{llll}9\hfill & 4\hfill & 3\hfill & -6\hfill \\ 2\hfill & -1\hfill & 7\hfill & 5\hfill \end{array}\right]$

Now, the transpose of the matrix D is,

${\left[D\right]}^T==\left[\begin{array}{cc}9& 2\\ 4& -1\\ 3& 7\\ -6& 5\end{array}\right]$.

Therefore, ${\left[D\right]}^T==\left[\begin{array}{cc}9& 2\\ 4& -1\\ 3& 7\\ -6& 5\end{array}\right]$.

(9) Consider the matrices A and C,

$\left[A\right]=\left[\begin{array}{ll}4\hfill & 7\hfill \\ 1\hfill & 2\hfill \\ 5\hfill & 6\hfill \end{array}\right]$ and $\left[C\right]=\left\{\begin{array}{l}3\hfill \\ 6\hfill \\ 1\hfill \end{array}\right\}$.

Since, interior dimensions of $\left[A\right]\times \left\{C\right\}$ are not equals. Hence, $\left[A\right]\times \left\{C\right\}$ is not possible.

Therefore, $\left[A\right]\times \left\{C\right\}$ is not possible.

(10) Consider the identity matrix I of dimension $3\times 3$ and matrix B,

$\left[B\right]=\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]$.

Since, the multiplication of the identity matrix with any matrix is again that matrix.

Hence,

$\begin{array}{c}\left[I\right]\times \left[B\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\times \left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]\\ =\left[\begin{array}{ccc}4& 3& 7\\ 1& 2& 7\\ 2& 0& 4\end{array}\right]\end{array}$

Therefore, $\left[I\right]\times \left[B\right]=\left[B\right]$.

(11) Consider the matrix E,

$\left[\text{E}\right]=\left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]$.

Now the transpose of the matrix E is,

${\left[\text{E}\right]}^T=\left[\begin{array}{ccc}1& 7& 4\\ 5& 2& 0\\ 8& 3& 6\end{array}\right]$.

The multiplication of the matrix E and its transpose is,

$\begin{array}{c}{\left[\text{E}\right]}^T\times \left[\text{E}\right]=\left[\begin{array}{ccc}1& 7& 4\\ 5& 2& 0\\ 8& 3& 6\end{array}\right]\times \left[\begin{array}{ccc}1& 5& 8\\ 7& 2& 3\\ 4& 0& 6\end{array}\right]\\ =\left[\begin{array}{ccc}1\times 1+7\times 7+4\times 4& 1\times 5+7\times 2+4\times 0& 1\times 8+7\times 3+4\times 6\\ 5\times 1+2\times 7+0\times 4& 5\times 5+2\times 2+0\times 0& 5\times 8+2\times 3+0\times 6\\ 8\times 1+3\times 7+6\times 4& 8\times 5+3\times 2+6\times 0& 8\times 8+3\times 3+6\times 6\end{array}\right]\\ =\left[\begin{array}{ccc}66& 19& 53\\ 19& 29& 46\\ 53& 46& 109\end{array}\right]\end{array}$

Therefore, ${\left[\text{E}\right]}^T\times \left[\text{E}\right]=\left[\begin{array}{ccc}66& 19& 53\\ 19& 29& 46\\ 53& 46& 109\end{array}\right]$.

(12) Consider the matrix C,

$\left[C\right]=\left\{\begin{array}{l}3\hfill \\ 6\hfill \\ 1\hfill \end{array}\right\}$

Now the transpose of the matrix C is,

${\left[C\right]}^T=\left\{\begin{array}{ccc}3& 6& 1\end{array}\right\}$

Multiplication of the matrix C and its transpose is,

$\begin{array}{c}{\left[C\right]}^T\times \left[C\right]=\left\{\begin{array}{ccc}3& 6& 1\end{array}\right\}\times \left\{\begin{array}{l}3\hfill \\ 6\hfill \\ 1\hfill \end{array}\right\}\\ =\left[3\times 3+6\times 6+1\times 1\right]\\ =\left[46\right]\end{array}$

Therefore, ${\left[C\right]}^T\times \left[C\right]=\left[46\right]$.