Evaluate the following:
Form an expression for the determinant of the matrix \begin{bmatrix} -5n & 2n^2 \\ -7 & 9n\end{bmatrix}.
If B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22}\end{bmatrix} is the inverse of A=\begin{bmatrix} -8 & -9\pi \\ \dfrac{1}{3} & \dfrac{1}{4}\end{bmatrix}, find:
\text{det}(A)
b_{11}
b_{12}
b_{21}
b_{22}
Consider the equation \begin{vmatrix} 9 & 6 \\ 3 & n\end{vmatrix}= 18. Solve for n.
For each matrix A:
Determine the inverse of the following matrices:
State the condition for a matrix to have an inverse.
Hence determine if the matrix \begin{bmatrix} 4 & 2 \\ -5 & 6\end{bmatrix} has an inverse.
The following matrices are inverses:
X= \begin{bmatrix} 7 & 4 \\ 5 & 3\end{bmatrix} \text{ and } Y=\begin{bmatrix} 3 & -4 \\ -5 & 7\end{bmatrix}
Calculate XY.
Calculate YX.
For each pair of matrices:
Find AB.
Hence state whether A and B are inverses of each other.
Let A = \begin{bmatrix} \dfrac {9}{2} & -\dfrac {5}{2} \\ \\9 & -\dfrac {9}{4}\end{bmatrix} and B=\begin{bmatrix} -2 & -1 \\ -\dfrac {7}{4} & -\dfrac {1}{3} \end{bmatrix}.
Find A^{ - 1 }.
Find X, if AX=B.
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}
Let A,B and C be matrices. Using matrix algebra, solve for matrix B in the following equations:
Find the matrix X for the given matrices and equations:
P = \begin{bmatrix} 13 & 3 \\ 7 & 11\end{bmatrix}, PX = P
N = \begin{bmatrix} 2 & -9 \\ -1 & -4\end{bmatrix} and I=\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}, X N = I
M = \begin{bmatrix} 1 & -5 \\ -5 & 6\end{bmatrix} and N=\begin{bmatrix} 7 & 8 \\ 9 & 1\end{bmatrix}, XM = N
M = \begin{bmatrix} 10 & -1 \\ -9 & 4\end{bmatrix} and N=\begin{bmatrix} 7 & -6 \\ 5 & -9\end{bmatrix}, M X = N
M = \begin{bmatrix} -6 & 10 \\ -6 & -4\end{bmatrix} and P=\begin{bmatrix} -6 & -10 \\ 4 & 1\end{bmatrix}, X P = M
M = \begin{bmatrix} 16 & -10 \\ 16 & 2\end{bmatrix} and N=\begin{bmatrix} -6 & -12 \\ -18 & -3\end{bmatrix}, M X = N
A = \begin{bmatrix} 10 & 3 \\8& 4 \end{bmatrix} and P=\begin{bmatrix} 5 \\ 6 \end{bmatrix}, A X = P
B = \begin{bmatrix} -2 & -3 \\ 3 & -10 \end{bmatrix} and Q=\begin{bmatrix} 5 \\ 6 \end{bmatrix}, B X = Q
A = \begin{bmatrix} -8 & 5 & 7 \\ 5 & 6 & -8 \\ 7 & 2 & 7 \end{bmatrix} and B=\begin{bmatrix} -10 \\ 1 \\ -8 \end{bmatrix}, A X = B
M = \begin{bmatrix} -2 & -8 \\ -1 & -6.6\end{bmatrix} and N=\begin{bmatrix} 5.4 & 1.6 \\ 4 & -3.1\end{bmatrix}, M X = N
M = \begin{bmatrix} -3 & -5 \\ 3 & 0\end{bmatrix}, N=\begin{bmatrix} 5 & 2 \\ 0 & -2 \end{bmatrix} and P=\begin{bmatrix} 6 & 10 \\ 10 & -9 \end{bmatrix}, MNX = P
M = \begin{bmatrix} 3 & 1 \\ -4 & 5\end{bmatrix}, N=\begin{bmatrix} 2 & -3 \\ 3 & -5 \end{bmatrix} and P=\begin{bmatrix} -6 & 10 \\ 5 & 5 \end{bmatrix}, M^{ - 1 } N X = P