Educational

Find Matrices X and Y Finding Matrices X and Y Question: If \[ 2X – Y = \begin{bmatrix}6 & -6 & 0 \\ -4 & 2 & 1\end{bmatrix}, \quad X + 2Y = \begin{bmatrix}3 & 2 & 5 \\ -2 & 1 & -7\end{bmatrix} \] find matrices \(X\) and \(Y\). Solution: Step 1: Eliminate \(Y\) […]

Find Matrix X Finding Matrix X Question: Given \[ Y=\begin{bmatrix}3 & 2 \\ 1 & 4\end{bmatrix} \] and \[ 2X + Y = \begin{bmatrix}1 & 0 \\ -3 & 2\end{bmatrix} \] Find \(X\). Solution: Step 1: Rearrange the equation \[ 2X = \begin{bmatrix}1 & 0 \\ -3 & 2\end{bmatrix} – \begin{bmatrix}3 & 2 \\ 1

Find Matrices X and Y Finding Matrices X and Y Question: If \[ X+Y=\begin{bmatrix}5 & 2 \\ 0 & 9\end{bmatrix}, \quad X-Y=\begin{bmatrix}3 & 6 \\ 0 & -1\end{bmatrix} \] find matrices \(X\) and \(Y\). Solution: Step 1: Add the equations \[ (X+Y)+(X-Y)=2X \] \[ = \begin{bmatrix}5 & 2 \\ 0 & 9\end{bmatrix} + \begin{bmatrix}3 &

Associative Property of Matrix Addition Verification of Associative Property Question: Given \[ A=\begin{bmatrix}2&1&1\\3&-1&0\\0&2&4\end{bmatrix},\; B=\begin{bmatrix}9&7&-1\\3&5&4\\2&1&6\end{bmatrix},\; C=\begin{bmatrix}2&-4&3\\1&-1&0\\9&4&5\end{bmatrix} \] Verify that: \[ (A+B)+C = A+(B+C) \] Solution: Step 1: Compute \(A+B\) \[ A+B= \begin{bmatrix} 2+9 & 1+7 & 1+(-1)\\ 3+3 & -1+5 & 0+4\\ 0+2 & 2+1 & 4+6 \end{bmatrix} = \begin{bmatrix} 11 & 8 & 0\\ 6

Diagonal Matrix Operations Diagonal Matrix Operations Question: Let \[ A = \text{diag}(2,-5,9), \quad B = \text{diag}(1,1,-4), \quad C = \text{diag}(-6,3,4) \] Find: (i) \(A – 2B\) (ii) \(B + C – 2A\) (iii) \(2A + 3B – 5C\) Solution: For diagonal matrices, operations are performed on corresponding diagonal elements. (i) \(A – 2B\) \[ A

Matrix Operation 2A – 3B + 4C Matrix Operation: \(2A – 3B + 4C\) Question: Let \[ A=\begin{bmatrix}-1 & 0 & 2 \\ 3 & 1 & 4\end{bmatrix}, \quad B=\begin{bmatrix}0 & -2 & 5 \\ 1 & -3 & 1\end{bmatrix}, \quad C=\begin{bmatrix}1 & -5 & 2 \\ 6 & 0 & -4\end{bmatrix} \] Compute \(2A

Matrix Operations (Defined & Not Defined) Matrix Operations (Defined / Not Defined) Question: Let \[ A=\begin{bmatrix}2 & 3 \\ 5 & 7\end{bmatrix}, \quad B=\begin{bmatrix}-1 & 0 & 2 \\ 3 & 4 & 1\end{bmatrix}, \quad C=\begin{bmatrix}-1 & 2 & 3 \\ 2 & 1 & 0\end{bmatrix} \] Find: (i) \(A+B\), \(B+C\) (ii) \(2B+3A\), \(3C-4B\) Solution:

Matrix Operations Example Matrix Operations Question: Let \[ A=\begin{bmatrix}2 & 4 \\ 3 & 2\end{bmatrix}, \quad B=\begin{bmatrix}1 & 3 \\ -2 & 5\end{bmatrix}, \quad C=\begin{bmatrix}-2 & 5 \\ 3 & 4\end{bmatrix} \] Find: (i) \(2A – 3B\) (ii) \(B – 4C\) (iii) \(3A – C\) (iv) \(3A – 2B + 3C\) Solution: (i) \(2A –

Matrix Addition of 3×3 Matrices Matrix Addition (3×3) Example Question: Compute: \[ \begin{bmatrix} 2 & 1 & 3 \\ 0 & 3 & 5 \\ -1 & 2 & 5 \end{bmatrix} + \begin{bmatrix} 1 & -2 & 3 \\ 2 & 6 & 1 \\ 0 & -3 & 1 \end{bmatrix} \] Solution: Add corresponding

Find a and b by Equating Matrices Finding a and b by Equating Matrices Question: Find \( a \) and \( b \) if \[ A = \begin{bmatrix} a+4 & 3b \\ 8 & -6 \end{bmatrix}, \quad B = \begin{bmatrix} 2a+2 & b^2+2 \\ 8 & b^2-5b \end{bmatrix} \] Concept Used Two matrices are equal