A =

⎛ 2   1 ⎞
⎝ 1   1 ⎠

 and

B =

⎛ 2  −5 ⎞
⎝ 0    1 ⎠

If AX + 3B = 0, then the determinant of X is:

(A) −18
(B) −6
(C) 6
(D) 18

Determinant of Matrix X When AX + 3B = 0: Detailed Matrix Algebra Solution

Correct Option: (D) 18

Understanding the Given Matrix Equation

We are given the matrix equation AX + 3B = 0, where A and B are known 2 × 2 matrices and X is an unknown matrix. The question asks only for the determinant of matrix X. Therefore, instead of calculating every individual element of X, we can solve the problem efficiently by applying the important properties of determinants, inverse matrices, and scalar multiplication.

Starting with the given matrix equation,

AX + 3B = 0

Moving 3B to the right-hand side gives:

AX = −3B

To isolate X, multiply both sides from the left by A⁻¹. Since A⁻¹A = I, where I represents the identity matrix, we obtain:

A⁻¹AX = −3A⁻¹B

X = −3A⁻¹B

This expression allows us to calculate the determinant of X directly by using standard determinant properties.

Finding the Determinant of Matrix X

From the equation

X = −3A⁻¹B,

taking determinants on both sides gives:

det(X) = det(−3A⁻¹B)

For an n × n matrix M, the determinant of a scalar multiple follows the property:

det(kM) = kⁿdet(M)

In this problem, the matrices are 2 × 2 matrices. Therefore, n = 2, and the scalar −3 contributes a factor of:

(−3)² = 9

Hence,

det(X) = 9 det(A⁻¹B)

Using the determinant property det(PQ) = det(P)det(Q), we get:

det(X) = 9 det(A⁻¹)det(B)

We also know that:

det(A⁻¹) = 1 / det(A)

Therefore,

det(X) = 9 × det(B) / det(A)

Calculating the Determinant of Matrix A

The given matrix A is:

A =

⎛ 2   1 ⎞
⎝ 1   1 ⎠

For a 2 × 2 matrix, the determinant is calculated by multiplying the elements of the main diagonal and subtracting the product of the other diagonal. Therefore,

det(A) = (2 × 1) − (1 × 1)

det(A) = 2 − 1

det(A) = 1

Since det(A) is non-zero, matrix A is invertible. Therefore, the use of A⁻¹ in the solution is mathematically valid.

Calculating the Determinant of Matrix B

The given matrix B is:

B =

⎛ 2  −5 ⎞
⎝ 0    1 ⎠

Applying the determinant formula,

det(B) = (2 × 1) − ((−5) × 0)

det(B) = 2 − 0

det(B) = 2

Substituting the Determinant Values

We obtained the formula:

det(X) = 9 × det(B) / det(A)

Substituting det(A) = 1 and det(B) = 2 gives:

det(X) = 9 × 2 / 1

det(X) = 18

Therefore, the determinant of matrix X is 18, making Option (D) the correct answer.

Detailed Explanation of Every Option

Option (A): −18

Option (A) is incorrect. The value −18 may be obtained if the negative sign in the scalar −3 is incorrectly retained after taking the determinant. Since X is a 2 × 2 matrix, the scalar contributes (−3)² = 9. The square removes the negative sign, so the final determinant must be positive.

Option (B): −6

Option (B) is incorrect. This answer may result from multiplying the determinant by −3 only once. However, for a 2 × 2 matrix, multiplying the matrix by a scalar k multiplies its determinant by k². Therefore, the effect of −3 must be squared.

Option (C): 6

Option (C) is incorrect. The value 6 can be obtained by calculating 3 × det(B) = 3 × 2 while ignoring the rule det(kM) = kⁿdet(M). Since the matrices are of order 2, the correct scalar factor is 3² = 9, not 3.

Option (D): 18

Option (D) is correct. From X = −3A⁻¹B, we obtain det(X) = (−3)²det(A⁻¹)det(B). Since det(A) = 1, det(A⁻¹) = 1, and det(B) = 2, the final calculation is 9 × 1 × 2 = 18.

Final Answer

The given matrix equation AX + 3B = 0 can be rewritten as X = −3A⁻¹B. Applying determinant properties gives det(X) = (−3)² × det(B) / det(A). Since det(A) = 1 and det(B) = 2, we obtain:

det(X) = 9 × 2 = 18

Correct Option: (D) 18