Given that Q = [[2, 1, 3], [2, 1, 2], [1, 2, 1]], find |Q|. Evaluate A⁻¹ x A.

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ScanSolve AI Tutor · Accepted Answer

Here's how to find the determinant of matrix $Q$ :

Write down the given matrix $Q$ .
$Q = \begin{pmatrix} 2 & 3 & 1 \\ 3 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix}$
Use the cofactor expansion method to find the determinant $|Q|$.
For a $3 \times 3$ matrix $\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$ , the determinant is given by $a(ei - fh) - b(di - fg) + c(dh - eg)$ .
Applying this formula to matrix $Q$ :
$|Q| = 2 \begin{vmatrix} 1 & 2 \\ 2 & 1 \end{vmatrix} - 3 \begin{vmatrix} 3 & 2 \\ 2 & 1 \end{vmatrix} + 1 \begin{vmatrix} 3 & 1 \\ 2 & 2 \end{vmatrix}$
Calculate the determinants of the $2 \times 2$ sub-matrices.
- For the first term:
$\begin{vmatrix} 1 & 2 \\ 2 & 1 \end{vmatrix} = (1 \times 1) - (2 \times 2) = 1 - 4 = -3$
- For the second term:
$\begin{vmatrix} 3 & 2 \\ 2 & 1 \end{vmatrix} = (3 \times 1) - (2 \times 2) = 3 - 4 = -1$
- For the third term:
$\begin{vmatrix} 3 & 1 \\ 2 & 2 \end{vmatrix} = (3 \times 2) - (1 \times 2) = 6 - 2 = 4$
Substitute these values back into the expression for $|Q|$ and simplify.
$|Q| = 2(-3) - 3(-1) + 1(4)$ $|Q| = -6 + 3 + 4$ $|Q| = -6 + 7$ $|Q| = \boxed{1}$
Part (ii) of the question is incomplete in the image provided.

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