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3. Matrix Inversion

Matrix Inversion

Ex 1.

Find the inverse matrix using the formula for a \(2\times2\) matrix

\[ A=\begin{pmatrix}2 & 1\\ 5 & 3\end{pmatrix} \qquad B=\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix} \qquad C=\begin{pmatrix}4 & 7\\ 2 & 6\end{pmatrix} \]

Ex 2.

For the matrices

\[ A=\begin{pmatrix}1 & 2\\ 2 & 5\end{pmatrix} \quad B=\begin{pmatrix}12& 5\\ 7 & 3\end{pmatrix} \quad C=\begin{pmatrix}1 & 2 & 3\\ 0 & 1 & 4\\ 5 & 6 & 0\end{pmatrix} \quad D=\begin{pmatrix}2 & 0 & 1\\ 1 & 3 & 0\\ 0 & 4 & 5\end{pmatrix} \]

calculate the inverse matrices using the methods:

  • augmenting with the identity matrix and performing Gauss-Jordan elimination,
  • using the formula with cofactor matrices (adjugate matrix)

So for each matrix provide two methods of calculating the inverse matrix (if it exists).

Ex 3.

Check if the matrix

\[ H=\begin{pmatrix}1 & 2 & 3\\ 2 & 4 & 6\\ 0 & 1 & 1\end{pmatrix} \]

is invertible. Justify the answer (use the determinant). Could this have been noticed without calculating the determinant? What would have to happen for the matrix to be invertible?

Ex 4.

For a matrix \(A\) satisfying \(A^{2}=I\) (so-called involution), show that \(A^{-1}=A\). Give an example of a non-trivial \(2\times2\) matrix satisfying this condition (other than \(I\) and \(-I\)). How many such matrices are there?

Ex 5.

Calculate the inverse of the diagonal matrix \(D=\operatorname{diag}(2,5,-3,1)\), if it exists. Discuss the condition for the existence of an inverse for a diagonal matrix.

Ex 6.

Solve the matrix equations:

a)

\[\begin{bmatrix} 2 & 5 \\\ 1 & 3 \end{bmatrix} \cdot X = \begin{bmatrix} 4 & -6 \\\ 2 & 1 \end{bmatrix}\]

b)

\[\begin{bmatrix} 2 & 1 \\\ 5 & 3 \end{bmatrix} \cdot X = \begin{bmatrix} 1 & 2 \\\ 3 & 4 \end{bmatrix}\]

c)

\[X \cdot \begin{bmatrix} 1 & 1 & -1 \\\ 2 & 1 & 0 \\\ 1 & -1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & -3 & 3 \\\ 4 & 3 & 2 \\\ 1 & -2 & 5 \end{bmatrix}\]

d)

\[\begin{bmatrix} 3 & 2 & 3 \\\ 1 & 1 & 2 \\\ 3 & 2 & 4 \end{bmatrix} \cdot X = \begin{bmatrix} 1 & 2 & 3 \\\ 1 & -1 & 2 \\\ 2 & 2 & 4 \end{bmatrix}\]