Problem 73 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Choose a candidate for the self-inverse matrix

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

Step 2: 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.