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Given that the determinant of matrix M = [[x-3, -1], [x+2, 3]] is 5, find the value of x.

July 13, 2026Mathematics

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Write down the formula for the determinant of a $2 \times 2$ matrix.
For a matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ , the determinant is given by $\det(A) = ad - bc$ .
Apply the determinant formula to the given matrix $M$ .
The matrix is $M = \begin{pmatrix} x-3 & -1 \\ x+2 & 3 \end{pmatrix}$ .
Here, $a = x-3$ , $b = -1$ , $c = x+2$ , and $d = 3$ .
$\det(M) = (x-3)(3) - (-1)(x+2)$
Set the determinant equal to 5 and solve for $x$ .
$(x-3)(3) - (-1)(x+2) = 5$ $3x - 9 + (x+2) = 5$ $3x - 9 + x + 2 = 5$ $4x - 7 = 5$ $4x = 5 + 7$ $4x = 12$ $x = \frac{12}{4}$ $x = \boxed{3}$

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