Class CBSE Class 12 Mathematics Matrices and Determinants Q #1808
COMPETENCY BASED
APPLY
5 Marks 2026 AISSCE(Board Exam) LA
If $P=\begin{bmatrix}1&-1&0\\2&3&4\\0&1&2\end{bmatrix}$ and $Q=\begin{bmatrix}2&2&-4\\-4&2&-4\\2&-1&5\end{bmatrix}$, find (QP) and hence solve the following system of equations using matrices: $x-y=3$, $2x+3y+4z=17$, $y+2z=7$
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Detailed Solution

Step 1: Calculate the product QP

We perform matrix multiplication of Q and P: $$QP = \begin{bmatrix}2&2&-4\\-4&2&-4\\2&-1&5\end{bmatrix} \begin{bmatrix}1&-1&0\\2&3&4\\0&1&2\end{bmatrix}$$ Calculating each element: Row 1: (2*1+2*2-4*0)=6, (2*-1+2*3-4*1)=0, (2*0+2*4-4*2)=0 Row 2: (-4*1+2*2-4*0)=0, (-4*-1+2*3-4*1)=6, (-4*0+2*4-4*2)=0 Row 3: (2*1-1*2+5*0)=0, (2*-1-1*3+5*1)=0, (2*0-1*4+5*2)=6 Thus, $$QP = 6I$$, where I is the identity matrix.

Step 2: Express the system as a matrix equation

The system is $PX = B$, where: $$P = \begin{bmatrix}1&-1&0\\2&3&4\\0&1&2\end{bmatrix}, X = \begin{bmatrix}x\\y\\z\end{bmatrix}, B = \begin{bmatrix}3\\17\\7\end{bmatrix}$$

Step 3: Solve for X

Since $$QP = 6I$$, we have $$P^{-1} = \frac{1}{6}Q$$. $$X = P^{-1}B = \frac{1}{6}Q B$$ $$X = \frac{1}{6} \begin{bmatrix}2&2&-4\\-4&2&-4\\2&-1&5\end{bmatrix} \begin{bmatrix}3\\17\\7\end{bmatrix}$$ $$X = \frac{1}{6} \begin{bmatrix}6+34-28\\-12+34-28\\6-17+35\end{bmatrix} = \frac{1}{6} \begin{bmatrix}12\\-6\\24\end{bmatrix} = \begin{bmatrix}2\\-1\\4\end{bmatrix}$$

Final Answer: x=2, y=-1, z=4

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to utilize the result of a matrix product to derive the inverse of a matrix and subsequently solve a linear system.
Knowledge Dimension: PROCEDURAL
Justification: The question tests the specific algorithmic process of matrix inversion via product properties rather than rote memorization of formulas.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This aligns with the NEP 2020 focus on testing conceptual understanding of matrix algebra and its application in solving systems of linear equations.

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