Educational

2×2 Matrix Both Symmetric and Skew-Symmetric 📘 Question Write a \(2 \times 2\) matrix which is both symmetric and skew-symmetric. ✏️ Step-by-Step Solution Step 1: Recall definitions Symmetric matrix: \(A^T = A\) Skew-symmetric matrix: \(A^T = -A\) Step 2: Combine both conditions If a matrix is both symmetric and skew-symmetric, then: \[ A^T = A […]

Find a – 2b Using Matrix Equality 📘 Question Solve the matrix equation: \[ \begin{bmatrix}a + 4 & 3b \\ 8 & -6\end{bmatrix} = \begin{bmatrix}2a + 2 & b + 2 \\ 8 & a – 8b\end{bmatrix} \] Find the value of \(a – 2b\). ✏️ Step-by-Step Solution Step 1: Compare corresponding elements \(a +

Find x in Matrix Equation 📘 Question Solve the matrix equation: \[ \begin{bmatrix}x & 1\end{bmatrix} \begin{bmatrix}1 & 0 \\ -2 & 0\end{bmatrix} = O \] Find the value of \(x\), where \(O\) is the zero matrix. ✏️ Step-by-Step Solution Step 1: Perform matrix multiplication \[ \begin{bmatrix}x & 1\end{bmatrix} \begin{bmatrix}1 & 0 \\ -2 & 0\end{bmatrix}

Find x – y in Matrix Equation 📘 Question Solve the matrix equation: \[ 2\begin{bmatrix}3 & 4 \\ 5 & x\end{bmatrix} + \begin{bmatrix}1 & y \\ 0 & 1\end{bmatrix} = \begin{bmatrix}7 & 0 \\ 10 & 5\end{bmatrix} \] Find the value of \(x – y\). ✏️ Step-by-Step Solution Step 1: Multiply the matrix by 2

If A² = A, Find 7A – (I + A)³ 📘 Question If a square matrix \( A \) satisfies \( A^2 = A \), find the value of: \[ 7A – (I + A)^3 \] where \( I \) is the identity matrix. ✏️ Step-by-Step Solution Given: \[ A^2 = A \] This means

Find p such that A^2 = pA Find p such that A2 = pA Given: \[ A = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix} \] Step 1: Compute A2 \[ A^2 = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix} \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix} \] \[

Find x for Skew-Symmetric Matrix Find x such that Matrix is Skew-Symmetric Given: \[ A = \begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & 3 \\ x & -3 & 0 \end{bmatrix} \] Condition: \[ A^T = -A \] Step 1: Compare Elements \[ a_{13} = -a_{31} \Rightarrow -2 = -x \]

Find b from Matrix Equation Find b Given: \[ \begin{bmatrix} a – b & 2a + c \\ 2a – b & 3c + d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix} \] Step 1: Form Equations \[ a – b = -1 \quad (1) \] \[ 2a – b =

Find Matrix A Find Matrix A Given: \[ \begin{bmatrix} 9 & -1 & 4 \\ -2 & 1 & 3 \end{bmatrix} = A + \begin{bmatrix} 1 & 2 & -1 \\ 0 & 4 & 9 \end{bmatrix} \] Step 1: Rearrange \[ A = \begin{bmatrix} 9 & -1 & 4 \\ -2 & 1 &

Find x from Vector Equation Find x Given: \[ x\begin{bmatrix} 2 \\ 3 \end{bmatrix} + y\begin{bmatrix} -1 \\ 1 \end{bmatrix} = \begin{bmatrix} 10 \\ 5 \end{bmatrix} \] Step 1: Form Equations \[ 2x – y = 10 \] \[ 3x + y = 5 \] Step 2: Add Equations \[ (2x – y) + (3x