Solving
Given the matrix \( A = \begin{bmatrix} 3 & 2 \\ 4 & 1 \end{bmatrix} \), calculate the inverse of matrix A.
Simplifying Matrices
Given matrices A = \( \begin{bmatrix} 2 & 3\\ 4 & -1 \end{bmatrix} \) and B = \( \begin{bmatrix} 1 & 5\\ -2 & 3 \end{bmatrix} \), find the matrix (2A - 3B).
Solving the System of Equations Using an Inverse Matrix
Given matrices A = \(\begin{bmatrix} 2 & 3 \\ 4 & 7 \end{bmatrix}\) and B = \(\begin{bmatrix} 1 \\ 2 \end{bmatrix}\), solve for X in the system of equations AX = B using the inverse matrix method.
Finding the Dimensions
What are the dimensions of the matrix \( A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \end{pmatrix} \)?
Multiplication by a Scalar
If \( A = \begin{bmatrix} 2 & 4\\ 1 & 3\end{bmatrix} \), find the result of the scalar multiplication \( 5A \).
Multiplication
If \( A = \begin{bmatrix} 1 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix} 2 \\ 4 \end{bmatrix} \), what is the product of \( A \) and \( B \)?
Subtraction
Given two 3x3 matrices, A = \(\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}\) and B = \(\begin{bmatrix} 9 & 8 & 7 \\ 6 & 5 & 4 \\ 3 & 2 & 1 \end{bmatrix}\), what is the result of A - B?
Finding the Determinant of the Resulting Matrix
Find the determinant of the matrix \[ A = \left[ \begin{array}{ccc} 2 & 3 & 4 \\ 1 & 5 & 6 \\ 7 & 8 & 9 \end{array} \right] \]
Finding the Inverse of the Resulting Matrix
Find the inverse of the matrix \[ A = \begin{pmatrix} 2 & 3 \ 4 & 5 \end{pmatrix} \]
Finding the Identity Matrix
Given the matrix \( A = \begin{bmatrix} 3 & 4 \\ 2 & 3 \end{bmatrix} \), find the matrix \( B \) such that \( AB = I \), where \( I \) is the identity matrix.
Finding the Scalar multiplied by the Identity Matrix
Find the result of the scalar 5 multiplied by the 2x2 identity matrix.
Addition
If Matrix A is \(\begin{bmatrix}2 & 3\ \ 4 & 5\ \ 6 & 7\end{bmatrix}\) and Matrix B is \(\begin{bmatrix}1 & 2\ \ 3 & 4\ \ 5 & 6\end{bmatrix}\), what is A + B?
Finding the Determinant of a 2x2 Matrix
Find the determinant of the 2x2 matrix \( \begin{bmatrix} 4 & 2 \\ 3 & 5 \end{bmatrix} \).
Finding the Determinant of a 3x3 Matrix
Find the determinant of the 3x3 matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \)
Finding the Determinant of Large Matrices
Find the determinant of the matrix \n\[ A = \begin{bmatrix} 5 & 2 & 1 & 3 \\ 4 & 3 & 2 & 1 \\ 1 & 2 & 3 & 4 \\ 3 & 1 & 2 & 5 \end{bmatrix} \]
Inverse of a 2x2 Matrix
Find the inverse of the matrix \( A = \begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix} \)
Finding the Transpose
Find the transpose of the matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \)
Finding the Adjoint
Find the adjoint of the matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix} \).
Finding the Cofactor Matrix
Find the cofactor matrix of the following 3x3 matrix: \[A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{bmatrix}\]
Null Space
Find the null space of the matrix \( A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix} \)
Finding the Basis and Dimension for the Column Space of the Matrix
Find the basis and dimension for the column space of the matrix \(A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}\)
Finding the Basis and Dimension for the Row Space of the Matrix
Find the basis and dimension for the row space of the matrix \[A=\begin{bmatrix} 2 & 4 & -2 \cr 1 & 2 & -1 \cr 3 & 6 & -3 \end{bmatrix}\]
Finding the LU Decomposition of a Matrix
Find the LU Decomposition of the matrix \( A = \begin{bmatrix} 4 & 3 \newline 6 & 3 \end{bmatrix} \)