May 2, 2024
In this lecture, we learned how to perform operations on matrices including matrix multiplication, scalar multiplication, and operations involving the identity matrix. We applied these concepts to evaluate expressions like ( A^2 - 3A + I ) and ( A^2 - 5I - 2A ), where ( A ) is a matrix and ( I ) is the identity matrix.
Matrix ( A ): [ A = \begin{bmatrix} 2 & 3 \ 1 & 5 \end{bmatrix} ]
Matrix Squared ( A^2 ): [ A^2 = \begin{bmatrix} 2 & 3 \ 1 & 5 \end{bmatrix} \times \begin{bmatrix} 2 & 3 \ 1 & 5 \end{bmatrix} = \begin{bmatrix} 7 & 21 \ 7 & 28 \end{bmatrix} ]
Scalar Multiplication ( 3A ): [ 3A = 3 \times \begin{bmatrix} 2 & 3 \ 1 & 5 \end{bmatrix} = \begin{bmatrix} 6 & 9 \ 3 & 15 \end{bmatrix} ]
Identity Matrix ( I ): [ I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} ]
Final Expression Evaluation: [ A^2 - 3A + I = \begin{bmatrix} 7 & 21 \ 7 & 28 \end{bmatrix} - \begin{bmatrix} 6 & 9 \ 3 & 15 \end{bmatrix} + \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 12 \ 4 & 14 \end{bmatrix} ]
Matrix ( A ): [ A = \begin{bmatrix} 1 & -3 \ 4 & 2 \end{bmatrix} ]
Matrix Squared ( A^2 ): [ A^2 = \begin{bmatrix} 1 & -3 \ 4 & 2 \end{bmatrix} \times \begin{bmatrix} 1 & -3 \ 4 & 2 \end{bmatrix} = \begin{bmatrix} -11 & -9 \ 12 & -8 \end{bmatrix} ]
Scalar Multiplicated Identity Matrix ( 5I ): [ 5I = 5 \times \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 0 \ 0 & 5 \end{bmatrix} ]
Scalar Multiplicated Matrix ( 2A ): [ 2A = 2 \times \begin{bmatrix} 1 & -3 \ 4 & 2 \end{bmatrix} = \begin{bmatrix} 2 & -6 \ 8 & 4 \end{bmatrix} ]
Final Expression Evaluation: [ A^2 - 5I - 2A = \begin{bmatrix} -11 & -9 \ 12 & -8 \end{bmatrix} - \begin{bmatrix} 5 & 0 \ 0 & 5 \end{bmatrix} - \begin{bmatrix} 2 & -6 \ 8 & 4 \end{bmatrix} = \begin{bmatrix} -18 & -3 \ 4 & -17 \end{bmatrix} ]
These examples demonstrate how to handle complex mathematical operations with matrices, showing the step-by-step process needed for solving each part of the expressions involving various types of matrix manipulations.