Why (A+B)(A-B) ≠ A² − B² Question If \(A\) and \(B\) are square matrices of the same order, explain why in general \[ (A + B)(A – B) \ne A^2 – B^2. \] Solution Step 1: Expand \[ (A + B)(A – B) \] \[ = A(A – B) + B(A – B) \] \[ […]
Why (A-B)² ≠ A² − 2AB + B² Question If \(A\) and \(B\) are square matrices of the same order, explain why in general \[ (A – B)^2 \ne A^2 – 2AB + B^2. \] Solution Step 1: Expand \((A-B)^2\) \[ (A – B)^2 = (A – B)(A – B) \] \[ = A^2 –
Why (A+B)² ≠ A² + 2AB + B² Question If \(A\) and \(B\) are square matrices of the same order, explain why in general \[ (A + B)^2 \ne A^2 + 2AB + B^2. \] Solution Step 1: Expand \((A+B)^2\) \[ (A + B)^2 = (A + B)(A + B) \] \[ = A^2 +
Matrix Identity (A+B)² Question Let \(A\) and \(B\) be square matrices of the same order. Does \[ (A + B)^2 = A^2 + 2AB + B^2 \] hold? If not, why? Solution Step 1: Expand \((A+B)^2\) \[ (A + B)^2 = (A + B)(A + B) \] \[ = A^2 + AB + BA +
Example AB = AC but B ≠ C Question Give an example of matrices \(A, B, C\) such that \[ AB = AC \quad \text{but} \quad B \ne C,\ A \ne O. \] Solution Step 1: Take Matrices \[ A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 1
Give example of matrix : A, B and C such that AB = AC but B ≠ C, A ≠ O. Read More »
Example AB = O but BA ≠ O Question Give an example of matrices \(A\) and \(B\) such that \[ AB = O \quad \text{but} \quad BA \ne O. \] Solution Step 1: Take Matrices \[ A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 0 \\
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Example AB = O but A ≠ O, B ≠ O Question Give an example of matrices \(A\) and \(B\) such that \[ AB = O \quad \text{but} \quad A \ne O,\ B \ne O. \] Solution Step 1: Take Matrices \[ A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}, \quad B
Give example of matrix : A and B such that AB = O but A ≠ O, B ≠ O. Read More »
Example AB ≠ BA Question Give an example of matrices \(A\) and \(B\) such that \[ AB \ne BA. \] Solution Step 1: Take Matrices \[ A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \] Step 2: Compute \(AB\) \[
Give example of matrix : A and B such that AB ≠ BA. Read More »
Matrix Dimension Problem Question Matrix \(X\) has \((a+b)\) rows and \((a+2)\) columns. Matrix \(Y\) has \((b+1)\) rows and \((a+3)\) columns. Both \(XY\) and \(YX\) exist. Find \(a\) and \(b\). Also determine whether \(XY\) and \(YX\) are of same type and equal. Solution Step 1: Condition for \(XY\) to exist \[ \text{Columns of } X =
Transpose Power Identity Question If \(A\) is a square matrix, prove using mathematical induction that \[ (A^T)^n = (A^n)^T \quad \forall n \in \mathbb{N}. \] Solution (Mathematical Induction) Step 1: Base Case (n = 1) \[ (A^T)^1 = A^T \quad \text{and} \quad (A^1)^T = A^T \] ✔ True for \(n=1\) Step 2: Assume for \(n