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

If \( [a_{ij}] = 2i - j \), then determine a matrix A of order \( 2 \times 3 \).
If \( A = \begin{bmatrix} 0 & -\tan{\frac{\alpha}{2}} \\[4pt] \tan{\frac{\alpha}{2}} & 0 \end{bmatrix} \), then prove that \[ (I + A) = (I - A)\begin{bmatrix} \cos{\alpha} & -\sin{\alpha} \\[4pt] \sin{\alpha} & \cos{\alpha} \end{bmatrix}. \]
Find the inverse of the matrix \( A = \begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix} \).
For two invertible matrices A and B of order n, prove that \( (AB)^{-1} = B^{-1}A^{-1} \).
If \(A = \begin{bmatrix} 2 & -3 & 5
3 & 2 & -4
1 & 1 & -2 \end{bmatrix}\), find \(A^{-1}\). Using \(A^{-1}\) solve the following system of equations:
2x - 3y + 5z = 11
3x + 2y - 4z = -5
x + y - 2z = -3.
If \(A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}\), then show that \(A^2 - 5A + 7I = O\). Using this, obtain \(A^{-1}\).
If \(A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}\), then find \(A^{-1}\).
If \(A = \begin{bmatrix} 3 & \sqrt{3} & 2 \\ 4 & 2 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 0 & 1/4 \\ 0 & 0 \\ 1/2 & 1/8 \end{bmatrix}\), then prove that \(|C| = 1\), where \(C = (A')' B\).
If \( A \) and \( B \) are two matrices of order \( n \) which are invertible, then prove that \( (AB)^{-1} = B^{-1}A^{-1} \).
For matrix \( A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \end{bmatrix} \) show that \( A^3 - 6A^2 + 5A + 11I = 0 \) and with the help of this find \( A^{-1} \).
If the matrices \(A = \begin{bmatrix} 3 & \sqrt{3} & 2 \\ 4 & 2 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 0 & 1/4 \\ 0 & 0 \\ 1/2 & 1/8 \end{bmatrix}\), then prove that \((A')' B = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\).
If \( F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} \), prove that \(F(x)F(y) = F(x+y)\).
If \(A = \begin{bmatrix} 0 & -\tan(\alpha/2) \\ \tan(\alpha/2) & 0 \end{bmatrix}\) and \(I\) is the identity matrix of order 2, prove that \(I+A = (I-A)\begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}\).
If \( 2X + Y = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix} \) and \( Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix} \), then \( X \) will be
If \(A' = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix}\) and \(B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}\), find \((A+2B)'\).
If \( A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix} \), verify that \( A'A=I \).
Solve the following system of equations by matrix method :
2x + 3y + 3z = 5
x - 2y + z = -4
3x - y - 2z = 3
Solve the following system of equations by matrix method:
\(x + y + 2z = 1\)
\(3x + 2y + z = 7\)
\(2x + y + 3z = 2\)
Solve by matrix method the following system of equations:
\(2x + y + z = 1\)
\(x - 2y - z = \frac{3}{2}\)
\(3y - 5z = 9\)
If the orders of the matrices A and B are \(m \times n\) and \(n \times p\) respectively, then the order of AB will be
If the matrix A = \(\begin{bmatrix} 0 & 2y & z
x & y & -z
x & -y & z \end{bmatrix}\) satisfies the equation AA' = I, then find the values of x, y and z.
If \( A = \begin{bmatrix} 3 & 3 & 1 \\ 3 & 4 & 1 \\ 4 & 3 & 1 \end{bmatrix} \), then verify that \( A \, (\text{adj } A) = |A| I \) and find \( A^{-1} \).
If \( A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix} \), then verify that \( A \cdot \text{adj}(A) = |A| I \) and find \( A^{-1} \).
Express the matrix \( A = \begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix} \) as the sum of a symmetric and a skew-symmetric matrix.
If \( \begin{bmatrix} x+y & 2 \\ 5+z & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix} \), then find the values of \(x, y, z\).