Lab 2

Exercise 1 (5 points) Prove that any square matrix \bold{A} can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

Exercise 2 (5 points) Prove that if a matrix is both symmetric and skew-symmetric, it must be a zero matrix.

Exercise 3 (5 points) If \begin{bmatrix}3x & 3\end{bmatrix}\begin{bmatrix}1 & 2\\ -4 & 0\end{bmatrix}\begin{bmatrix}x & 8\end{bmatrix}^T=\bold{0}, find the value of x.

Exercise 4 (5 points)  

Write the following system of linear equations in matrix notation. \begin{aligned} 2x_1 - x_3 &= 0\\ x_2 - 5x_1 &= -1\\ x_3 + 6x_2 &= 2. \end{aligned}

Exercise 5 (5 points) Consider a n \times n matrix \bold{A}. Show that if \bold{A} is not invertible, then there is a n \times n matrix \bold{B} with \bold{AB}=\bold{0} but \bold{B} \ne \bold{0}.

Exercise 6 (5 points) Give an example of two matrices that can not be multiplied?

Exercise 7 (5 points) Give an example of two 2\times2 matrices \bold{A} and \bold{B} such that \bold{AB}\neq\bold{BA}?

Exercise 8 (5 points) If \bold{A}=\begin{bmatrix}-1 & 0 & 2\\0 & 5 & 2\\-4 & 0 & -3\end{bmatrix} and \bold{I} denotes the 3\times3 identity matrix, calculate the expression 6103\bold{I}^{2025} - \bold{A}^2.

Exercise 9 (5 points) Find the matrix \bold{A} that satisfies \frac{3}{2}\bold{A}+\begin{bmatrix}1 & 0 & -2\\-1 & 2 & 2\\4 & 1 & 2\end{bmatrix}=\begin{bmatrix}0 & 1 & 2\\1 & 0 & 3\\0 & 2 & -2\end{bmatrix}.

Exercise 10 (5 points) Find determinant of \bold{A}=\begin{bmatrix}10 & -3\\1 & 2\end{bmatrix}.

Exercise 11 (5 points) Find the inverse of \bold{A}=\begin{bmatrix}2 & 3\\-1 & 0\end{bmatrix}, provided it exists.

Exercise 12 (5 points) What value must x have, so the matrix \bold{A}=\begin{bmatrix}1 & 2+x\\ x & -1\end{bmatrix} does not have an inverse?