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Chapter 6: Problem 73

Give an example of a \(2 \times 2\) matrix that is its own inverse.

Short Answer

Expert verified
One example of a \(2 \times 2\) matrix that is its own inverse is \(A = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}\)

Step by step solution

01

Choose a candidate for the self-inverse matrix

Here, we will use a very simple matrix that is likely to be its own inverse: \[A = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}\]
02

Confirm that the chosen matrix is the self-inverse

Now we will multiply our chosen matrix \(A\) by itself and confirm whether the product equals the identity matrix \(I\) by performing the matrix multiplication: \[ \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix} \times \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix} = \begin{bmatrix} 1*1+0*0 & 1*0+0*(-1) \ 0*1+(-1)*0 & 0*0+(-1)*(-1) \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \] The result is the identity matrix, confirming that our candidate matrix \(A\) is indeed its own inverse.

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Most popular questions from this chapter

In Exercises \(9-16,\) find: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{ll} 4 & 1 \\ 3 & 2 \end{array}\right], \quad B=\left[\begin{array}{ll} 5 & 9 \\ 0 & 7 \end{array}\right] $$

In Exercises \(27-36,\) find (if possible): \(\begin{array}{llll}\text { a. } A B & \text { and } & \text { b. } B A\end{array}\) $$ A=\left[\begin{array}{l} -1 \\ -2 \\ -3 \end{array}\right], \quad B=\left[\begin{array}{lll} 1 & 2 & 3 \end{array}\right] $$

In Exercises \(5-8,\) find values for the variables so that the matrices in each exercise are equal. $$ \left[\begin{array}{rr} x & 2 y \\ z & 9 \end{array}\right]=\left[\begin{array}{rr} 4 & 12 \\ 3 & 9 \end{array}\right] $$

In Exercises \(37-44\), perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} 4 & 0 \\ -3 & 5 \\ 0 & 1 \end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\ -2 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right] $$ $$ B C+C B $$

In Exercises \(17-26,\) let $$ A=\left[\begin{array}{rr} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{array}\right] \text { and } B=\left[\begin{array}{rr} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{array}\right] $$ Solve each matrix equation for \(X\). $$ X-B=A $$
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