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Identify the given matrix A.

May 3, 2026Mathematics

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22

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Identify the given matrix A.
The matrix A is a $3 \times 3$ matrix:
$A = \begin{vmatrix} -2 & 3 & 1 \\ 2 & -1 & 3 \\ 1 & 2 & 1 \end{vmatrix}$
Calculate the determinant using cofactor expansion along the first row.
The formula for the determinant of a $3 \times 3$ matrix $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}$ is $a(ei - fh) - b(di - fg) + c(dh - eg)$ .
Substitute the values from matrix A into the determinant formula.
$\det(A) = -2 \begin{vmatrix} -1 & 3 \\ 2 & 1 \end{vmatrix} - 3 \begin{vmatrix} 2 & 3 \\ 1 & 1 \end{vmatrix} + 1 \begin{vmatrix} 2 & -1 \\ 1 & 2 \end{vmatrix}$
Calculate the $2 \times 2$ determinants.
$\begin{vmatrix} -1 & 3 \\ 2 & 1 \end{vmatrix} = (-1)(1) - (3)(2) = -1 - 6 = -7$ $\begin{vmatrix} 2 & 3 \\ 1 & 1 \end{vmatrix} = (2)(1) - (3)(1) = 2 - 3 = -1$ $\begin{vmatrix} 2 & -1 \\ 1 & 2 \end{vmatrix} = (2)(2) - (-1)(1) = 4 - (-1) = 4 + 1 = 5$
Substitute these values back into the expression for $\det(A)$ and simplify.
$\det(A) = -2(-7) - 3(-1) + 1(5)$ $\det(A) = 14 + 3 + 5$ $\det(A) = 22$
The final answer is 22.

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