Given the matrix A = \begin{pmatrix} 3 & 1 \\ 4 & 2 \end{pmatrix}, what is the determinant… — JAMB UTME

Given the matrix $A = \begin{pmatrix} 3 & 1 \\ 4 & 2 \end{pmatrix}$ , what is the determinant of the matrix $3A$ ?

The determinant of the matrix

3A

18

For any

2 \times 2

matrix, scaling the matrix by a factor

k

scales its determinant by

k^2

. Since

\det(A) = (3)(2) - (1)(4) = 2

, multiplying the matrix by

3

yields

\det(3A) = 3^2 \times 2 = 9 \times 2 = 18

. Alternatively, computing

3A = \begin{pmatrix} 9 & 3 \\ 12 & 6 \end{pmatrix}

directly gives

\det(3A) = (9)(6) - (3)(12) = 54 - 36 = 18

Calculate the determinant of matrix

A

\det(A) = (3)(2) - (1)(4) = 6 - 4 = 2.

The determinant of a

2 \times 2

matrix

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

is given by

ad - bc

Apply the determinant scalar multiplication property for an

n \times n

matrix.

\det(3A) = 3^2 \det(A) = 9 \times 2 = 18.

For any

n \times n

square matrix

A

and scalar

k

, the identity

\det(kA) = k^n \det(A)

holds. Here,

k=3

and

n=2

Scalar Multiplication Property of Determinants

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