Chapter 11.4, Problem 5AYU

To determine

To fill: The blank in the statement “Suppose that A is a square n by n matrix that is nonsingular. The matrix B such that $AB=BA=I_n$ is called of the matrix A.”

Answer to Problem 5AYU

The complete statement is “Suppose that A is a square n by n matrix that is nonsingular. The matrix B such that $AB=BA=I_n$ is called inverse of the matrix A.”

Explanation of Solution

Given information:

The statement “Suppose that A is a square n by n matrix that is nonsingular. The matrix B such that $AB=BA=I_n$ is called of the matrix A.”

Consider the provided statement “Suppose that A is a square n by n matrix that is nonsingular. The matrix B such that $AB=BA=I_n$ is called of the matrix A.”

When the two matrices are multiplied and follow commutative law of multiplication then one matrix is the inverse of other.

For example: Let $A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]$ .

Now,

  $\begin{array}{c}AB=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\end{array}$

And,

  $\begin{array}{c}BA=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\end{array}$

Therefore, $AB=BA=I_2$ .

So, $A=B^{-1}$

Thus, the complete statement is “Suppose that A is a square n by n matrix that is nonsingular. The matrix B such that $AB=BA=I_n$ is called inverse of the matrix A.”