April 2026

Show AB ≠ BA Show That \(AB \ne BA\) Question: Given \[ A=\begin{bmatrix}5 & -1 \\ 6 & 7\end{bmatrix}, \quad B=\begin{bmatrix}2 & 1 \\ 3 & 4\end{bmatrix} \] show that \(AB \ne BA\). Solution: Step 1: Compute \(AB\) \[ AB = \begin{bmatrix} 5(2)+(-1)(3) & 5(1)+(-1)(4) \\ 6(2)+7(3) & 6(1)+7(4) \end{bmatrix} \] \[ = \begin{bmatrix} 10-3 […]

Matrix Multiplication (3×3 × 3×3) Matrix Multiplication (3×3) Question: Compute: \[ \begin{bmatrix} 2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{bmatrix} \begin{bmatrix} 1 & -3 & 5 \\ 0 & 2 & 4 \\ 3 & 0 & 5 \end{bmatrix} \] Solution: Multiply rows of first

Matrix Multiplication (2×2 × 2×3) Matrix Multiplication Question: Compute: \[ \begin{bmatrix} 1 & -2 \\ 2 & 3 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ -3 & 2 & -1 \end{bmatrix} \] Solution: Multiply rows of first matrix with columns of second matrix: \[ = \begin{bmatrix} 1(1) + (-2)(-3) & 1(2) + (-2)(2) &

Matrix Multiplication Identity Matrix Multiplication Question: Compute: \[ \begin{bmatrix} a & b \\ -b & a \end{bmatrix} \begin{bmatrix} a & -b \\ b & a \end{bmatrix} \] Solution: Multiply row by column: \[ = \begin{bmatrix} a\cdot a + b\cdot b & a(-b) + b\cdot a \\ -b\cdot a + a\cdot b & (-b)(-b) + a\cdot

Income Expenditure Matrix Problem Income & Expenditure (Matrix Method) Question: The monthly incomes of Aryan and Babban are in the ratio 3:4 and their expenditures are in the ratio 5:7. If each saves ₹15000 per month, find their monthly income using matrix method. Also state the value reflected. Solution: Step 1: Assume incomes \[ \text{Aryan’s

Column Matrix and Scalar Multiplication Column Matrix & Scalar Multiplication Question: In a city, there are 30 colleges. Each college has: 15 peons 6 clerks 1 typist 1 section officer Express this as a column matrix and find total posts using scalar multiplication. Solution: Step 1: Column Matrix Representation \[ A = \begin{bmatrix} 15 \\

Solve Matrix Equations for X and Y Solving for X and Y Question: If \[ 2X + 3Y = \begin{bmatrix}2 & 3 \\ 4 & 0\end{bmatrix}, \quad 3X + 2Y = \begin{bmatrix}-2 & 2 \\ 1 & -5\end{bmatrix} \] find matrices \(X\) and \(Y\). Solution: Step 1: Eliminate Y Multiply first equation by 2 and

Find x and y from Matrix Equation Finding x and y Question: Solve: \[ 2\begin{bmatrix}x & 5 \\ 7 & y-3\end{bmatrix} + \begin{bmatrix}3 & 4 \\ 1 & 2\end{bmatrix} = \begin{bmatrix}7 & 14 \\ 15 & 14\end{bmatrix} \] Solution: Step 1: Multiply \[ 2\begin{bmatrix}x & 5 \\ 7 & y-3\end{bmatrix} = \begin{bmatrix}2x & 10 \\

Find x, y, z, t from Matrix Equation Finding x, y, z, t Question: Solve: \[ 3\begin{bmatrix}x & y \\ z & t\end{bmatrix} = \begin{bmatrix}x & 6 \\ -1 & 2t\end{bmatrix} + \begin{bmatrix}4 & x+y \\ z+t & 3\end{bmatrix} \] Solution: Step 1: LHS \[ = \begin{bmatrix} 3x & 3y \\ 3z & 3t \end{bmatrix}

Find Matrix X (3×2) Finding Matrix X (3×2) Question: If \[ A=\begin{bmatrix}8 & 0 \\ 4 & -2 \\ 3 & 6\end{bmatrix}, \quad B=\begin{bmatrix}2 & -2 \\ 4 & 2 \\ -5 & 1\end{bmatrix} \] find matrix \(X\) of order \(3\times2\) such that: \[ 2A + 3X = 5B \] Solution: Step 1: Rearrange the