Evaluate the following:
State the condition for a matrix to have an inverse.
Hence, determine if each of the following matrices has an inverse:
Form an expression for the determinant of the matrix \begin{bmatrix} -5n & 2n^2 \\ -7 & 9n\end{bmatrix}.
Consider the equation \begin{vmatrix} 9 & 6 \\ 3 & n\end{vmatrix}= 18. Solve for n.
If B=\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} is the inverse of A=\begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}, find:
the determinant of A.
the value of each of the following:
b_{11}
b_{12}
b_{21}
b_{22}
For each matrix A:
For each matrix A:
Find the determinant of A. Give your answer in exact form.
Find A^{-1}.
A=\begin{bmatrix} -8 & -9\pi \\\\ \dfrac{1}{3} & \dfrac{1}{4}\end{bmatrix}
Determine the inverse of each of the following matrices. Round each element to two decimal places if necessary.
Given the following pairs of matrices are inverses of each other, state their product:
\begin{bmatrix} 7 & 4 \\ 5 & 3 \end{bmatrix} and \begin{bmatrix} 3 & -4 \\ -5 & 7 \end{bmatrix}
\begin{bmatrix} 2 & 3 & 7 \\ 6 & 0 & 5 \\ 1 & 4 & 8 \end{bmatrix} and \begin{bmatrix} 20 & -4 & -15 \\ 43 & -9 & -32 \\ -24 & 5 & 18 \end{bmatrix}
For each pair of matrices:
Find AB.
Hence, state whether A and B are inverses of each other.
A=\begin{bmatrix} 7 & 2 \\ 3 & 1 \end{bmatrix} and B=\begin{bmatrix} 1 & -2 \\ -3 & 7 \end{bmatrix}
A=\begin{bmatrix} 7 & 3 \\ 5 & 2 \end{bmatrix} and B=\begin{bmatrix} 2 & -3 \\ -5 & 7 \end{bmatrix}
A=\begin{bmatrix} 3 & 3 & 1 \\ 2 & 1 & 2 \\ 4 & 4 & 1 \end{bmatrix} and B=\begin{bmatrix} -7 & 1 & 5 \\ 6 & -1 & -4 \\ 4 & 0 & -3 \end{bmatrix}
A=\begin{bmatrix} 1 & 1 & 2 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} and B=\begin{bmatrix} 0 & 1 & 0 \\ 1 & -1 & -2 \\ 0 & 0 & 1 \end{bmatrix}
Consider A = \begin{bmatrix} -2 & 5 \\ -1 & 2\end{bmatrix} \text{ and } B=\begin{bmatrix} 2 & -5 \\ 1 & -2\end{bmatrix}.
Find AB.
Find A^{-1}.
Given the matrix A and its inverse A^{-1}, find n:
A = \begin{bmatrix} 7 & 2 \\ 8 & 1\end{bmatrix} and A^{-1} =\begin{bmatrix} -\dfrac {1}{9} & \dfrac{2} {9} \\\\ \dfrac {8}{9} & n \end{bmatrix}
A = \begin{bmatrix} -1 & 2\\ 3 & -9\end{bmatrix} and A^{-1}=\begin{bmatrix} -3 & n \\\\ -1 & -\dfrac{1}{3}\end{bmatrix}
A = \begin{bmatrix} 5 & n \\ -2 & -4\end{bmatrix} and A^{-1} =\begin{bmatrix} \dfrac {2}{3} & \dfrac{7}{6} \\\\ -\dfrac {1}{3} & -\dfrac{5}{6}\end{bmatrix}
A = \begin{bmatrix} 7 & 5\\ 3 & n\end{bmatrix} and A^{-1}=\begin{bmatrix} \dfrac{3}{16} & -\dfrac{5}{48} \\\\ -\dfrac{1}{16} & \dfrac{7}{48}\end{bmatrix}