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: $x-y=3$, $2x+3y+4z=17$, $y+2z=7$
Prev Next

AI Tutor Explanation

Powered by Gemini

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

AI generated content. Review strictly for academic accuracy.

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.

More from this Chapter

MCQ_SINGLE
If $A=\begin{bmatrix}\cos x&-\sin x\\\sin x&\cos x\end{bmatrix}$ and $A+A^{\prime}=I$, then the value of $x \in [0,\frac{\pi}{2}]$ is
LA
A furniture workshop produces three types of furniture chairs, tables and beds each day. On a particular day the total number of furniture pieces produced is 45. It was also found that production of beds exceeds that of chairs by 8, while the total production of beds and chairs together is twice the production of tables. Determine the units produced of each type of furniture, using matrix method.
MCQ_SINGLE
Find the matrix \(A^{2}\), where \(A=[a_{ij}]\) is a \(2\times2\) matrix whose elements are given by \(a_{ij}=\) maximum (i, j) - minimum (i, j):
LA
Solve the following system of equations, using matrices: $\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4$ $\frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1$ , $\frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2$ where x, y, $z\ne0$
MCQ_SINGLE
If \(A=[a_{ij}]\) is an identity matrix, then which of the following is true ?
View All Questions