MTH1004 Linear Algebra Practical Class 5

Question 1

Question

Find the inverse of the following $2\times 2$ matrices: $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right),B=\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)$ using the standard formula for the inverse of $2\times 2$ matrices: $A^{-1}=\frac{1}{ad-bc}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$.

Solution

…

Question 2

Question

Find the inverse of the matrix $B=\left(\begin{array}{ccc}3& 1& 2\\ -1& 1& 2\\ 1& 3& 0\end{array}\right)$ using Gauss-Jordan elimination and its rank.

Solution

…

Question 3

Question

Prove that: (a) If $A$ is invertible and $AB=O$, then $B=O$. (b) If $A$ is invertible and $AB=AC$, then $B=C$. (c) If $A$ satisfies the matrix equation $A^2-2A+I=O$, then $A$ is invertible and find its inverse. (d) If $A$ satisfies the matrix equation $A^3+\alpha A^2+\beta A-I=O$, then $A$ is invertible and find its inverse. (a) Compute the determinant of the following matrices: $A=\left(\begin{array}{ccc}2& -3& 2\\ 4& 1& 0\\ 0& 0& 6\end{array}\right),B=\left(\begin{array}{ccc}-2& 1& 0\\ 1& -2& 1\\ 0& 1& -2\end{array}\right)$. (b) Find the determinant of the matrix $A^n$.

Solution

…

Question 4

Question

Using the determinant properties (and not cofactor expansions), show that: $\det (P^{-1}AP)=\det A$ $\det (P^{T}AP)=\det A(\det P{)}^2$ (c) If $\det A=1$ and $ABAT=B^2$, then $\det B=0$ or 1.

Solution

…

Question 5

Question

Use Cramer’s rule to solve the following linear systems: $\{\begin{array}{l}2x+y-z=1\\ -x+2y+z=1\\ x-y+2z=1\end{array}$, $\{\begin{array}{l}x+ay+az=0\\ x-y+az=0\\ x+y+z=0\end{array}$ For which $a$ values the second system has unique solution?

Solution

…