Linear Algebra Practical Class, Week 23

Question One

Question

Find the inverse of the following $2\times 2$ matrices, using the standard formula:

$$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$$

Solution

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Question Two

Question

Find the inverse of the following matrix using Gauss-Jordan elimination and its rank:

$$B=\left[\begin{array}{ccc}3& 1& 2\\ -1& 1& 2\\ 2& 1& 3\end{array}\right]$$

Solution

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Question Three

Question

Prove that…

1. If $A$ is invertible and $AB=O$, the $B=O$.
2. If $A$ is invertible and $AB=AC$, then $B=C$.
3. If $A$ satisfies the matrix equation $A^2-2A+I=O$, then $A$ is invertible and find its inverse.
4. If $A$ satisfies the matrix equation $A^3+\alpha A^2+\beta A-I=O$, then $A$ is invertible and find its inverse.

Solution

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Question Four

Question

Compute the determinant of the following matrices, then find the determinant of the matrix $A^n$:

$$A=\left[\begin{array}{ccc}2& -3& 2\\ 4& 1& 0\\ 0& 0& 6\end{array}\right],B=\left[\begin{array}{ccc}-2& 1& 0\\ 1& -2& 1\\ 0& 1& -2\end{array}\right]$$

Solution

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Question Five

Question

Using the determinant properties (and not cofactor expansions), show that:

1. $\det (P^{-1}AP)=\det A$,
2. $\det (P^{T}AP)=\det A(\det P{)}^2$.,
3. If $\det A=1$ and $AB{A}^T=B^2$, then $\det B=\{\begin{array}{l}0\\ 1\end{array}$.

Solution

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Question Six

Question

Use Cramer’s rule to solve the following linear systems and then determine which $a$ values the second system has a unique solution:

$$\{\begin{array}{l}2x+y-z=1\\ -x+2y+z=1\\ x-y+2z=1\end{array},\{\begin{array}{l}x+ay+az=0\\ x-y+az=0\\ x+y+z=0\end{array}$$

Solution

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