Given the matrices A = \begin{pmatrix} x & 2 & 1 \\ 3 & 0 & 2 \\ 1 & 4 & 1 \end{pmatrix} a… — JAMB UTME

Given the matrices $A = \begin{pmatrix} x & 2 & 1 \\ 3 & 0 & 2 \\ 1 & 4 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & -1 \\ 5 & 3 \end{pmatrix}$ , find the value of $x$ if $\det(A) + \det(B) = 5$ .

Answer: 2

Answer

The value of x is 2.

Evaluating \det(B) gives 11, and cofactor expansion of \det(A) along the top row yields 10 - 8x. Summing these determinants gives (10 - 8x) + 11 = 5, which simplifies to 8x = 16, resulting in x = 2.

Step-by-Step Solution

Calculate the determinant of the 2x2 matrix B

\det(B) = 11

For a 2x2 matrix \begin{pmatrix} a & b \\ c & d \end{pmatrix}, the determinant is ad - bc. Thus, \det(B) = (2)(3) - (-1)(5) = 6 + 5 = 11.

Calculate the determinant of the 3x3 matrix A in terms of x

\det(A) = 10 - 8x

Expanding along the first row: \det(A) = x\begin{vmatrix} 0 & 2 \\ 4 & 1 \end{vmatrix} - 2\begin{vmatrix} 3 & 2 \\ 1 & 1 \end{vmatrix} + 1\begin{vmatrix} 3 & 0 \\ 1 & 4 \end{vmatrix} = x(0 - 8) - 2(3 - 2) + 1(12 - 0) = -8x - 2 + 12 = 10 - 8x.

Set up and solve the linear equation \det(A) + \det(B) = 5

x = 2

Substituting the computed determinants gives (10 - 8x) + 11 = 5, which simplifies to 21 - 8x = 5. Rearranging yields 8x = 16, so x = 2.

Key Concept

Determinants of 2x2 and 3x3 matrices and linear equation solving

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