Class CBSE Class 12 Mathematics Matrices and Determinants Q #889
COMPETENCY BASED
APPLY
5 Marks 2023 LA
32. Find the inverse of the matrix $A=\begin{bmatrix}1 & -1 & 2\\ 0 & 2 & -3\\ 3 & -2 & 4\end{bmatrix}$. Using the inverse, solve the system of linear equations $x-y+2z=1; 2y-3z=1; 3x-2y+4z=3$.
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Step-by-Step Solution

Step 1: Find the determinant of matrix A.

det(A) = 1(2*4 - (-3)*(-2)) - (-1)(0*4 - (-3)*3) + 2(0*(-2) - 2*3) = 1(8 - 6) + 1(0 + 9) + 2(0 - 6) = 2 + 9 - 12 = -1

Step 2: Find the matrix of cofactors.

Cofactor matrix = $\begin{bmatrix} (2*4 - (-3)*(-2)) & -(0*4 - (-3)*3) & (0*(-2) - 2*3)\\ -((-1)*4 - 2*(-2)) & (1*4 - 2*3) & -(1*(-2) - (-1)*3)\\ ((-1)*(-3) - 2*2) & -(1*(-3) - 0*2) & (1*2 - (-1)*0) \end{bmatrix}$ = $\begin{bmatrix} 2 & -9 & -6\\ 0 & -2 & -1\\ -1 & 3 & 2 \end{bmatrix}$

Step 3: Find the adjugate (transpose of the cofactor matrix).

adj(A) = $\begin{bmatrix}2 & 0 & -1\\ -9 & -2 & 3\\ -6 & -1 & 2\end{bmatrix}$

Step 4: Find the inverse of A.

$A^{-1} = \frac{1}{det(A)} * adj(A) = \frac{1}{-1} * \begin{bmatrix}2 & 0 & -1\\ -9 & -2 & 3\\ -6 & -1 & 2\end{bmatrix} = \begin{bmatrix}-2 & 0 & 1\\ 9 & 2 & -3\\ 6 & 1 & -2\end{bmatrix}$

Step 5: Express the system of equations in matrix form.

$\begin{bmatrix}1 & -1 & 2\\ 0 & 2 & -3\\ 3 & -2 & 4\end{bmatrix} \begin{bmatrix}x\\ y\\ z\end{bmatrix} = \begin{bmatrix}1\\ 1\\ 3\end{bmatrix}$

Step 6: Solve for X using the inverse.

$X = A^{-1}B = \begin{bmatrix}-2 & 0 & 1\\ 9 & 2 & -3\\ 6 & 1 & -2\end{bmatrix} \begin{bmatrix}1\\ 1\\ 3\end{bmatrix} = \begin{bmatrix}-2*1 + 0*1 + 1*3\\ 9*1 + 2*1 + (-3)*3\\ 6*1 + 1*1 + (-2)*3\end{bmatrix} = \begin{bmatrix}1\\ 2\\ 1\end{bmatrix}$

Step 7: State the solution.

x = 1, y = 2, z = 1

Correct Answer: x = 1, y = 2, z = 1

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires students to apply the concept of matrix inverses to solve a system of linear equations. This involves using a learned procedure in a new context.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a specific method (finding the inverse of a matrix) and then applying that inverse to solve a system of equations. This is a step-by-step process.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question assesses the student's ability to apply the concept of matrix inverses to solve a system of linear equations, which is a practical application of the theory.

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