Chapter 2, Problem 1SE

Assume that the matrices mentioned in the statements below have appropriate sizes. Mark each statement True or False. Justify each answer.

1. a. If A and B are m × n. then both AB^T and A^T B are defined.
2. b. If AB = C and C has2 columns, then A has 2 columns.
3. c. Left-multiplying a matrix B by a diagonal matrix A, with nonzero entries on the diagonal, scales the rows of B.
4. d. If BC = BD, then C = D.
5. e. If AC = 0, then either A = 0 or C = 0.
6. f. If ,A and B are n × n. then (A + B)(A − B) = A²− B².
7. g. An elementary n × n matrix has either n or n + 1 nonzero entries.
8. h. The transpose of an elementary matrix is an elementary matrix.
9. i. An elementary matrix must be square.
10. j. Every square matrix is a product of elementary matrices.
11. k. If A is a 3 × 3 matrix with three pivot positions, there exist elementary matrices E₁, …, E_p such that E_p … E₁A = 1.
12. l. If AB = 1, then A is invertible.
13. m. If A and B are square and invertible, then AB is invertible, and (AB)⁻¹ = A⁻¹ B⁻¹.
14. n. If AB = BA and if A is invertible, then A⁻¹ B = BA⁻¹.
15. ○.      If A is invertible and if r ≠ 0, then (rA)⁻¹ = rA^−l.
16. p. If A is 3 × 3 matrix and the equation Ax = $\left[\begin{array}{l}1\\ 0\\ 0\end{array}\right]$ has a unique solution, then A is invertible.

To determine

To mark:

The given statement, “If A and B are $m\times n$, then both $A{B}^T$ and $A^{T}B$ are defined” as true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Justification of statement:

The dimension of matrix A and B are $m\times n$.

The $B^T$ has many rows and A has many columns.

Therefore, $A{B}^T$ is defined.

To determine

To mark:

The given statement, “If $AB=C$ and C has 2 columns, then A has 2 columns” as true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement:

The matrix B contains 2 columns because matrix A has many columns andB has many rows.

To determine

To mark:

The given statement, “Left-multiplying a matrix B by a diagonal matrix A, with nonzero entries on the diagonal, scales the rows of B” as true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Justification of statement:

$\left(0,\dots ,d_i,\dots ,0\right)$ is ith row of matrix A.

Therefore, $\left(0,\dots ,d_i,\dots ,0\right)B$ is the ith row of matrix AB.

$\left(0,\dots ,d_i,\dots ,0\right)B$ represents $d_i$ times ith row of B.

To determine

To mark:

The given statement, “If $BC=BD$, then $C=D$” as true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement:

Consider that matrix B and matrix equation $B\text{x}=0$ have nonzero solutions.

Construct matrix C and D (which are not equal, $C\ne D$) and columns of $C-D$ satisfy the equation $B\text{x}=0$.

Then

$\begin{array}{l}B\left(C-D\right)=0\\ BC-BD=0\\ BC=BD\end{array}$

To determine

To mark:

The given statement, “If $AC=0$, then either $A=0$ or $C=0$” as true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement:

If $AC=0$, then either $A=0$ or $C=0$.

Consider matrix A and B as shown below.

$\begin{array}{l}A=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\\ C=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\end{array}$

Product of matrix $AC=0$.

To determine

To mark:

The given statement, “If A and B are $n\times n$ matrix, then $\left(A+B\right)\left(A-B\right)=A^2-B^2$” as true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement:

Consider $m\times n$ matrix A and B.

$\left(A+B\right)\left(A-B\right)=A^2-AB+BA-B^2$

The value of $\left(A+B\right)\left(A-B\right)=A^2-B^2$, if matrix A commutes with B.

To determine

To mark:

The given statement, “An elementary $n\times n$ matrix has either n or $n+1$ nonzero entries” as true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Justification of statement:

The replacement matrix of $n\times n$ has $n+1$ nonzero entries.

The interchange and scale matrices have n nonzero entries.

To determine

To mark:

The given statement, “The transpose of an elementary matrix is an elementary matrix” as true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Justification of statement:

An elementary matrix is the same type of the transpose of an elementary matrix.

To determine

To mark:

The given statement, “An elementary matrix must be square” as true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Justification of statement:

An $n\times n$ elementary matrix is obtained by a row operation on $I_n$.

To determine

To mark:

The given statement, “Every square matrix is a product of elementary matrices” as true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement:

If an elementary matrix is invertible, then the products of matrices are also invertible.

However, each square matrix is not invertible.

To determine

To mark:

The given statement “If A is a $3\times 3$ matrix with three pivot positions, there exist elementary matrices $E_1,\dots ,E_P$ such that $E_P\cdots E_{1}A=I$” as true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Justification of statement:

ConsiderA is $3\times 3$ matrix with 3 pivot positions, then matrixA’srow is equivalent to $I_3$.

To determine

To mark:

The given statement “If $AB=I$, then A is invertible” as true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement:

Matrix A is a square invertible matrix from equation $AB=I$.

To determine

To mark:

The given statement “If A and B are square and invertible, then AB is invertible, and ${\left(AB\right)}^{-1}=A^{-1}{B}^{-1}$” as true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement:

The product AB of the matrix is invertible.

The inverse of product AB, ${\left(AB\right)}^{-1}=B^{-1}{A}^{-1}$.

The product ${\left(AB\right)}^{-1}$ is always not equal to $A^{-1}{B}^{-1}$.

To determine

To mark:

The given statement “If $AB=BA$ and if A is invertible, then $A^{-1}B=B{A}^{-1}$” as true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Justification of statement:

The product of matrix is given below:

$AB=BA$ (1)

Multiply the left by $A^{-1}$ on both sides of Equation (1).

$\begin{array}{l}{A}^{-1}AB=A^{-1}BA\\ B=A^{-1}BA\end{array}$

Multiply the right by $A^{-1}$ on both sides of Equation (1).

$\begin{array}{l}B{A}^{-1}=A^{-1}BA{A}^{-1}\\ B{A}^{-1}=A^{-1}B\end{array}$

To determine

To mark:

The given statement “If A is invertible and if $r\ne 0$, then ${\left(rA\right)}^{-1}=r{A}^{-1}$” as true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement:

The proper equation is ${\left(rA\right)}^{-1}=r^{-1}{A}^{-1}$.

Multiply by rA.

$\begin{array}{c}\left(rA\right){\left(rA\right)}^{-1}=\left(r{r}^{-1}\right)\left(A{A}^{-1}\right)\\ =I\times I\\ =I\end{array}$

To determine

To mark:

The given statement “If A is a $3\times 3$ matrix and the equation $A\text{x}=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$ has a unique solution, then A is invertible” as true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Justification of statement:

The equation $A\text{x}=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$ has a unique solution.

No free variables are there in $A\text{x}=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$.

Therefore,matrix A must have pivot positions.

Hence, matrix A is invertible.