Question

Is x=[26-1-4] the inverse of a=[12-214-12]? Select the correct answers from the drop-down menus to correctly complete the statements. The product of the matrices ______ the identity matrix. Therefore, x and a ______ inverses of each other.

1) is
2) are
3) are not
4) is not

Answer 1

The product of the matrices is the identity matrix. Therefore, x and a are inverses of each other.

The correct option is 1 and 2.

Step-by-step explanation:

Matrix multiplication is crucial in determining if two matrices are inverses. Given matrices
`\(a\) and \(x\), where \(a = \begin{bmatrix} 12 & -2 \\ 14 & -12 \end{bmatrix}\) and \(x = \begin{bmatrix} 26 & -1 \\ -4 & 0 \end{bmatrix}\),`we compute their product ax.

`\[ax = \begin{bmatrix} 12 & -2 \\ 14 & -12 \end{bmatrix} * \begin{bmatrix} 26 & -1 \\ -4 & 0 \end{bmatrix} = \begin{bmatrix} 240 & -2 \\ 62 & -14 \end{bmatrix}\]`

Now, to check if the product is the identity matrix, we compare it with the 2x2 identity matrix
`\(I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\). As \(ax \\eq I_2\),`it implies that x is not the inverse of a. There might be a typographical error or misunderstanding in the given matrices.

Therefore, the correct answer is that the product of the matrices is not the identity matrix, and consequently, x and a are not inverses of each other. The choice "are not" aligns with this conclusion. It's essential to be cautious with matrix calculations, as even small errors can lead to significant differences in the results.

The correct option is 1 and 2.