Question

Consider the matrix et(A) = -2" alt="A_{4x4}, det(A) = -2" align="absmiddle" class="latex-formula">
Show that the following is true or false:


$det(adj(2A^{-1}) =-2^{9}$

Answer 1

We found a counterexample, so the statement is false.

Is the statement true?

Let's use the matrix:

$\left[\begin{array}{cccc}-2&0&0&0\\0&1&0&0\\0&0&1&0\\ 0&0&0&1 \end{array}\right]$

This is a 4x4 matrix with determinant equal to -2.

The inverse matrix is:

$\left[\begin{array}{cccc}1/2&0&0&0\\0&-1&0&0\\0&0&-1&0\\ 0&0&0&-1 \end{array}\right]$

If we multiply it by 2, we get:

$\left[\begin{array}{cccc}1&0&0&0\\0&-2&0&0\\0&0&-2&0\\ 0&0&0&-2 \end{array}\right]$

The adjoint of that is the original matrix, actually:

$\left[\begin{array}{cccc}-2&0&0&0\\0&1&0&0\\0&0&1&0\\ 0&0&0&1 \end{array}\right]$

Which we already know, has a determinant of -2.

So the statement is false, as we found a counterexample.

If you want to learn more about matrices:

brainly.com/question/11989522

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