Decompose Matrix into Symmetric and Skew-Symmetric Parts Find Symmetric and Skew-Symmetric Matrices X and Y Given: \[ A = \begin{bmatrix} 3 & 2 & 7 \\ 1 & 4 & 3 \\ -2 & 5 & 8 \end{bmatrix} \] Formula Used: \[ X = \frac{1}{2}(A + A^T), \quad Y = \frac{1}{2}(A – A^T) \] Step […]
Find x, y, z, t in Symmetric Matrix Find x, y, z, t if A is a Symmetric Matrix Given: \[ A = \begin{bmatrix} 5 & 2 & x \\ y & z & -3 \\ 4 & t & -7 \end{bmatrix} \] Condition for Symmetric Matrix: \[ A = A^T \] Step 1: Compare
Show A – A^T is Skew-Symmetric Show that A − AT is a Skew-Symmetric Matrix Given: \[ A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} \] Step 1: Find AT \[ A^T = \begin{bmatrix} 3 & 1 \\ -4 & -1 \end{bmatrix} \] Step 2: Compute A − AT \[ A –
If A = [[3, -4], [1, -1]], show that A – A^T is a skew-symmetric matrix. Read More »
Prove A – A^T is Skew-Symmetric Prove that A − AT is Skew-Symmetric Given: \[ A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \] Step 1: Find AT \[ A^T = \begin{bmatrix} 2 & 4 \\ 3 & 5 \end{bmatrix} \] Step 2: Compute A − AT \[ A – A^T =
If A=[[2, 3], [4, 5]], prove that A – A^T is skew-symmetric matrix. Read More »
Prove AA^T = I using Direction Cosines Prove that AAT = I Given: \[ A = \begin{bmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3 \end{bmatrix} \] where \((l_i, m_i, n_i)\) are direction cosines of three mutually perpendicular unit vectors. Properties Used: \[ l_i^2 + m_i^2
Verify A^T A = I₂ Verify that ATA = I2 Given: \[ A = \begin{bmatrix} \sin\alpha & \cos\alpha \\ -\cos\alpha & \sin\alpha \end{bmatrix} \] Step 1: Find AT \[ A^T = \begin{bmatrix} \sin\alpha & -\cos\alpha \\ \cos\alpha & \sin\alpha \end{bmatrix} \] Step 2: Compute ATA \[ A^T A = \begin{bmatrix} \sin\alpha & -\cos\alpha \\ \cos\alpha
If A = [[sin α, cos α], [-cos α, sin α]], verify that A^T A = I2 Read More »
Verify A^T A = I₂ Verify that ATA = I2 Given: \[ A = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix} \] Step 1: Find AT \[ A^T = \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix} \] Step 2: Compute ATA \[ A^T A = \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha
If A = [[cos α, sin α], [-sin α, cos α]], then verify that A^T A = I2. Read More »
Find A^T – B^T Find AT − BT Given: \[ A^T = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix} \] Step 1: Find BT \[ B^T = \begin{bmatrix} -1 & 1 \\ 2
If A^T = [[3, 4], [-1, 2], [0, 1]] and B = [[-1, 2, 1], [1, 2, 3]], find A^T – B^T Read More »
Find A^T – B^T Find AT − BT Given: \[ A^T = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix} \] Step 1: Find BT \[ B^T = \begin{bmatrix} -1 & 1 \\ 2
If A^T = [[3, 4], [-1, 2], [0, 1]] and B = [[-1, 2, 1], [1, 2, 3]], find A^T – B^T Read More »
Verify (AB)^T = B^T A^T Verify that (AB)T = BTAT Given: \[ A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 4 \\ 2 & 5 \end{bmatrix} \] To Verify: \[ (AB)^T = B^T A^T \] Step 1: Find AB \[ AB = \begin{bmatrix} 1 & 3