List of top Mathematics Questions on Matrices asked in UP Board XII

If \(A=\begin{bmatrix}1&2&3\\3&-2&1\\4&2&1\end{bmatrix}\), then show that \(A^3-23A-40I=0\).
If \(A=\begin{bmatrix}2&3\\1&-4\end{bmatrix},\ B=\begin{bmatrix}1&-2\\-1&3\end{bmatrix}\), then find \((AB)^{-1}\).
If \(A^2=A\), then simplify \((I+A)^2 - 7A\), where \(A\) is a square matrix.
If \(A\) is any matrix and \(K\) is any constant, then \((KA)'\) is:
Prove that the matrix \(A=\begin{bmatrix}2&-1\\3&4\end{bmatrix}\) satisfies the equation \(A^2-6A+11I=O\), where \(I\) is the identity matrix of order \(2\times2\) and \(O\) is the zero matrix of order \(2\times2\). With the help of it, find \(A^{-1}\).
If \(2\begin{bmatrix}x&5\\7&y-3\end{bmatrix}+\begin{bmatrix}3&-4\\1&2\end{bmatrix}=\begin{bmatrix}7&6\\15&10\end{bmatrix}\), find the values of \(x\) and \(y\).
If \(A=\begin{bmatrix}1&2&3\\2&4&5\\3&5&6\end{bmatrix}\), then find \(A^{-1}\).
Solve the following system of equations by using the Matrix Method: \[ x-y+2z=7 \] \[ 3x+4y-5z=-5 \] \[ 2x-y+3z=12 \]
If \(A=\begin{bmatrix}2&-3&5\\3&2&-4\\1&1&-2\end{bmatrix}\), then find \(A^{-1}\).
If \(A=\begin{bmatrix}3&-2\\4&-2\end{bmatrix}\) and \(I=\begin{bmatrix}1&0\\0&1\end{bmatrix}\) and \(A^2=KA-2I\), then find the value of \(K\).
If \(A=\begin{bmatrix}1&2&3\\2&3&1\end{bmatrix}\) and \(B=\begin{bmatrix}3&-1&3\\-1&0&2\end{bmatrix}\), then find the value of \((2A-B)\).
\(A=[a_{ij}]_{m\times n}\) is a square matrix if:
Solve the following system of equations by matrix method: \(3x-2y+3z=8,\ 2x+y-z=1,\ 4x-3y+2z=4\).
Express the matrix \(B=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}\) in the form of the sum of a symmetric matrix and a skew-symmetric matrix.
Simplify: \(\cos\theta\begin{bmatrix}\cos\theta & \sin\theta\\-\sin\theta & \cos\theta\end{bmatrix}+\sin\theta\begin{bmatrix}\sin\theta & -\cos\theta\\\cos\theta & \sin\theta\end{bmatrix}\).
Suppose that \(A=\begin{bmatrix}2 & -1\\3 & 4\end{bmatrix}\), \(B=\begin{bmatrix}5 & 2\\7 & 4\end{bmatrix}\), \(C=\begin{bmatrix}2 & 5\\3 & 8\end{bmatrix}\). Find the matrix \(D\) such that \(CD-AB=0\).
If \(A=\begin{bmatrix}1 & 2 & 3\\3 & -2 & 1\\4 & 2 & 1\end{bmatrix}\), then show that \(A^3-23A-40I=0\).