If the matrix A = \begin{pmatrix} 1 & 0 & 2 \\ 2 & k & 1 \\ 3 & 1 & 4 \end{pmatrix} is sin… — JAMB UTME

If the matrix $A = \begin{pmatrix} 1 & 0 & 2 \\ 2 & k & 1 \\ 3 & 1 & 4 \end{pmatrix}$ is singular, what is the value of $k$ ?

\frac{3}{2}

A matrix is singular if its determinant equals zero. Expanding the determinant of matrix

A

along its first row gives

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

. Setting

-2k + 3 = 0

yields

k = \frac{3}{2}

Apply the condition for a singular matrix

A matrix is singular when its determinant equals zero, so

\det(A) = 0

By definition, square matrices with zero determinant are singular.

Expand

\det(A)

along the first row

\det(A) = 1(4k - 1) - 0(8 - 3) + 2(2 - 3k) = 4k - 1 + 4 - 6k = -2k + 3

Laplace expansion along the first row simplifies computation due to the zero entry.

Solve for

k

-2k + 3 = 0 \implies 2k = 3 \implies k = \frac{3}{2}

Isolating the variable gives the required value of

k

Singular Matrix Condition and 3x3 Determinant Expansion

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