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Question #40322

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ matrix with real entries such that $\det(A) > 0$ . The matrix of minors of $A$ is given by $M = \begin{pmatrix} -24 & -20 & -5 \\ -18 & -15 & -4 \\ 5 & 4 & 1 \end{pmatrix}$ If the ordered triple $(x, y, z)$ satisfies the matrix equation $\begin{pmatrix} x & y & z \end{pmatrix} A = \begin{pmatrix} 1 & -1 & 2 \end{pmatrix}$ then the value of $x + y + z$ is:

Options

A.
$-2$
B.
$-22$
C.
$-12$
D.
$8$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Ordered Pair Equality|System of Linear Equations (Cramer/Inverse)|Minors and Cofactors Sign Convention

Why This Question Is Tricky

Common Mistakes

The question tests the student's understanding of the relationship between the matrix of minors, the matrix of cofactors, and the adjoint of a matrix in solving linear systems.

The primary trap lies in the Determinant Sign Convention.

Students frequently conflate the matrix of minors with the matrix of cofactors, ignoring the $(-1)^{i+j}$ checkerboard sign pattern required to form the adjoint matrix.

If a student directly transposes the matrix of minors $M$ to find the inverse, they will incorrectly calculate the solution vector as $(-14, -11, 3)$ , leading to the trap answer $-22$ .

Another common error is forgetting to transpose the cofactor matrix, leading to $-12$ .

The problem forces rigorous application of both the sign convention and the transpose operation to find the correct inverse and solve the ordered pair equality.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Ordered Pair Equality
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free