Simplifying Matrices
If \( A=\begin{pmatrix} 3 & 5 \\ -1 & 2 \end{pmatrix} \) and \( B=\begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix} \), find the matrix \( C \) such that \( C = 2A - 3B \).
Finding the Variables
2. Calcule os termos desconhecidos:
a) $\left(\begin{array}{cc}x & 3 \\ 5 & 2 y\end{array}\right)=\left(\begin{array}{cc}6 & 3 \\ 5 & 8\end{array}\right)$
Finding the Dimensions
What are the dimensions of the matrix \( A = \left[ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} \right] \)?
Multiplication by a Scalar
If \( A = \begin{bmatrix} 2 & 3\\ 4 & 5 \end{bmatrix} \), find the result of the scalar multiplication \( 3A \).
Multiplication
Given two matrices \( A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \), find the product \( AB \).
Subtraction
Given two 2x2 matrices A = \( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \) and B = \( \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \), what is the result of A - B?
Finding the Determinant of the Resulting Matrix
Find the determinant of the following matrix: \n\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]
Finding the Inverse of the Resulting Matrix
If matrix \( A = \begin{bmatrix} 1 & 2 \cr 3 & 4 \end{bmatrix} \), what is \( A^{-1} \), the inverse of matrix A?
Finding the Identity Matrix
If \( A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \), find the matrix \( B \) such that \( AB = I \) where \( I \) is the identity matrix.
Addition
Given two matrices, \( A = \begin{bmatrix} 3 & 4 \\ 2 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} -1 & 2 \\ 3 & -2 \end{bmatrix} \), find the result of \( A + B \).
Simplifying the Matrix Operation
Given matrices A = \( \begin{bmatrix} 2 & 3 \\ 4 & -1 \\ \end{bmatrix} \) and B = \( \begin{bmatrix} 1 & 2 \\ 3 & 1 \\ \end{bmatrix} \), compute the product AB.
Finding the Determinant of a 2x2 Matrix
Find the determinant of the following 2x2 matrix: \[ \begin{bmatrix} 5 & 3 \\ 2 & 4 \end{bmatrix} \]
Finding the Determinant of a 3x3 Matrix
Find the determinant of the 3x3 matrix: \[A = \begin{bmatrix} 2 & 5 & -3 \ 1 & -2 & 2 \ 0 & 5 & -1 \end{bmatrix}\]
Finding the Determinant of Large Matrices
Find the determinant of the following 4x4 matrix: \[A = \begin{pmatrix} 3 & 2 & 1 & 4 \\ 0 & 1 & 0 & 2 \\ 1 & 0 & 2 & 1 \\ 0 & 1 & 0 & 1 \end{pmatrix}\]
Inverse of a 2x2 Matrix
Find the inverse of the 2x2 matrix \(A = \begin{bmatrix} 3 & 4 \\ 2 & 1 \end{bmatrix}\).
Inverse of an nxn Matrix
Find the inverse of the matrix \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \)
Finding the Cofactor Matrix
Find the cofactor matrix of the given 3x3 matrix A = [[1, 2, 3], [0, 4, 5], [1, 0, 1]]
Null Space
Find the null space of the following matrix. \[ A = \begin{bmatrix} 1 & 2 & -1 \ 2 & 4 & -2 \ -1 & -2 & 1 \end{bmatrix} \]
Finding the Pivot Positions and Pivot Columns
Find the pivot positions and pivot columns of the following matrix, \[ \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]
Finding the Basis and Dimension for the Column Space of the Matrix
Find the basis and the dimension of the column space for the following 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
Given the matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix}$, find the basis and dimension for the row space of the matrix.