Question

Difficulty: MediumMatrices and Determinants

Given the matrix $A = \begin{pmatrix} 1 & 2 & 0 \\ 3 & 4 & 1 \\ 2 & 0 & k \end{pmatrix}$ , if $\det(A) = 10$ , what is the value of $k$ ?

Answer: -3

The value of

k

-3

Expanding the determinant along the first row gives

1(4k) - 2(3k - 2) + 0 = -2k + 4

. Setting

-2k + 4 = 10

leads directly to

-2k = 6

, giving

k = -3

Expand the determinant of matrix

A

along the first row.

\det(A) = 1(4k - 0) - 2(3k - 2) + 0 = -2k + 4

Using cofactor expansion along the top row to find the expression for the determinant.

Set the calculated determinant equal to the given value and solve for

k

-2k + 4 = 10 \implies -2k = 6 \implies k = -3

Equating the determinant algebraic expression to 10.

Determinant of a 3x3 Matrix

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