Chapter 11.4, Problem 7AYU

To determine

To explain: Whether the statement “The identity matrix has properties similar to those to real number 1.” is true or false.

Answer to Problem 7AYU

The statement “The identity matrix has properties similar to those to real number 1.” is true.

Explanation of Solution

Given information:

The statement “The identity matrix has properties similar to those to real number 1.”

Consider the provided statement “A function can have two different horizontal asymptotes.”

The statement is true because in matrix algebra the identity matrix serve as the multiplicative identity.

In real numbers any number multiplied by 1 gives the same number. That is, 1 is the multiplicative identity.

  $a\cdot 1=a$

Where a is any real number.

In matrix algebra, consider a matrix $A=\left[\begin{array}{cc}2& -9\\ 4& 3\end{array}\right]$ the corresponding identity matrix is $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ . Recall that to find the product AB of two matrices A and B , the number of column in matrix A must be equal to number of rows in matrixB .

  $\begin{array}{c}AI=\left[\begin{array}{cc}2& -9\\ 4& 3\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\\ =\left[\begin{array}{cc}2& -9\\ 4& 3\end{array}\right]\end{array}$

Thus, the statement “The identity matrix has properties similar to those to real number 1.” is true.