Linear Algebra

How do you express following square matrix as a sum of symmetric and a skew symmetric matrix? A=[8 3 2;-1 0 6;4 2 5] Answer : We shall express following…

Question : If A is a 3 by 3 non-singular matrix such that AA^T = A^TA and B = A^(-1)AT,then BB^T is (A)I+B    (B) I      (c) A+B …

Question : Is the set {(1, -2),(-1, 1)} linearly dependent or linearly independent? Answer : Linearly Dependent Set : A set S of elements or vectors is said to be…

Question : Is the set {(1, 0),(0, 1),(0, −1)} linearly dependent ? Which vector is not a linear combination of the other 2 vectors. Answer : Linearly Dependent Set :…

Question : what is linear system of equations? How do you solve them? Please Explain Solution : Linear Equation : A general linear equation has the form\[a_1x_1 + a_2x_2 +…

Question : what is Echelon Form of a Matrix? Find Echelon Form of the following Matrix Solution : Echelon Form of a Matrix A matrix is in row echelon form…

Question : what is rank of matrix? Find rank of the following matrix \[\left( {\begin{array}{*{20}{c}}1&{ 2}&{ 3}\\{ 2}&4&{ 6}\\{ 4}&{ 8}&12\end{array}}\right)\] Solution : Rank of Matrix : Rank of Matrix…

Question : what is determinant? Find determinant of the following matrices \[\left(\begin{array}{cc}2 & -1 \\3 & 1 \end{array}\right)\] \[\left(\begin{array}{ccc}1 & -1 & 0 \\1 & 3 & -1 \\3 &…

Question : what are symmetric and skew symmetric matrices ? Give Examples Solution : Symmetric Matrix A matrix \(A\) whose transpose \(tr(A)\) is same as as \(A\) is called symmetric.…

Question : what are triangular matrices ? Give Examples Solution : Triangular Matrix A square matrix \(A=(a_{i,j})_{1 \le i,j \le n}\) whose entries satisfy either \(a_{i,j} =0 \forall i\ge j\)…