$\left[\begin{array}{cc}5 & 4 \\ 2 & -1\end{array}\right] \cdot\left[\begin{array}{c}-4 \\ 3\end{array}\right]$
Answer (1)
Multiply the first row of the 2x2 matrix by the column vector: ( 5 × − 4 ) + ( 4 × 3 ) = − 8 .
Multiply the second row of the 2x2 matrix by the column vector: ( 2 × − 4 ) + ( − 1 × 3 ) = − 11 .
The resulting 2x1 matrix is [ − 8 − 11 ] .
Explanation
Understanding the Problem We are asked to multiply a 2x2 matrix by a 2x1 matrix (a column vector). Let's perform this matrix multiplication step by step.
Identifying the Matrices The given matrices are:
Matrix A (2x2): [ 5 4 2 − 1 ]
Matrix B (2x1): [ − 4 3 ]
We need to compute A * B.
Performing the Multiplication To multiply matrix A by matrix B, we perform the following calculations:
First element of the resulting matrix: (5 * -4) + (4 * 3) = -20 + 12 = -8
Second element of the resulting matrix: (2 * -4) + (-1 * 3) = -8 - 3 = -11
Resulting Matrix The resulting matrix (2x1) is: [ − 8 − 11 ]
Final Answer Therefore, the product of the given matrices is: [ − 8 − 11 ]
Examples
Matrix multiplication is used in various fields such as computer graphics, physics, and engineering. For example, in computer graphics, matrices are used to perform transformations on 3D models, such as rotation, scaling, and translation. Each vertex of the model is represented as a vector, and the transformation is applied by multiplying the vertex vector by a transformation matrix. This allows for efficient manipulation and rendering of 3D objects on the screen.
