Problem 74 - Algebra and Trigonometry Real Mathematics, Real People

Question

Operations with Matrices Perform the operations (a) using a graphing utility and (b) by hand algebraically. If it is not possible to perform the operation(s), state the reason. $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -2 \\ -1 & 1 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} -1 & 4 & -1 \\ -2 & -1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rr} 1 & 2 \\ -2 & 3 \\ 1 & 0 \end{array}\right], \quad c=-2, \text { and } d=-3 \end{aligned}$$ $$B+d A$$

Step-by-Step Solution

Answer

The result of the operation \(B + dA\) is \(\left[\begin{array}{rrr} -4 & -2 & 5 \ 1 & -4 & 0 \end{array}\right]\)

Step 1: Scalar and Matrix Multiplication

Firstly, we need to multiply matrix A by the scalar 'd' which is -3. The multiplication of a matrix by a scalar involves multiplying each element of the matrix by the scalar.\n\n\(dA = -3 * A = \left[\begin{array}{rrr} -3*1 & -3*2 & -3*(-2) \ -3*(-1) & -3*1 & -3*0 \end{array}\right] = \left[\begin{array}{rrr} -3 & -6 & 6 \ 3 & -3 & 0 \end{array}\right]\)

Step 2: Matrix Addition

Next, we need to add the result found above to matrix 'B'. The addition of two matrices involves adding the corresponding elements together.\n\n\(B + dA = B + -3A = \left[\begin{array}{rrr} -1 & 4 & -1 \ -2 & -1 & 0 \end{array}\right] + \left[\begin{array}{rrr} -3 & -6 & 6 \ 3 & -3 & 0 \end{array}\right] = \left[\begin{array}{rrr} -4 & -2 & 5 \ 1 & -4 & 0 \end{array}\right]\)

Key Concepts

Matrix Addition

Matrix addition is a straightforward and essential operation in linear algebra. It involves adding corresponding elements from two matrices to produce a new matrix. This operation can only be performed on matrices of the same dimensions. For instance, if both matrices are of the order 2x3, meaning they have two rows and three columns, you can add them together.

• Each element in the resulting matrix is obtained by adding the corresponding elements from the original matrices.


• For example: given matrices \( A = \left[ \begin{array}{cc} a_{11} & a_{12} \ a_{21} & a_{22} \end{array} \right] \) and \( B = \left[ \begin{array}{cc} b_{11} & b_{12} \ b_{21} & b_{22} \end{array} \right] \)


• The resulting matrix \( C = A + B \) will be \( \left[ \begin{array}{cc} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{array} \right]. \)

This simple addition is foundational and is used in solving larger systems of equations where matrices represent different variables.

Scalar Multiplication

Scalar multiplication is the process where each entry of a matrix is multiplied by a constant, known as a scalar. This operation transforms the entire matrix by maintaining its shape while altering its individual entries. Simply put:

• If you have a matrix \( A = \left[ \begin{array}{cc} 1 & 2 \ 3 & 4 \end{array} \right] \) and a scalar \( c = 3 \), then \( cA \) would result in:


• \( 3A = \left[ \begin{array}{cc} 3 \times 1 & 3 \times 2 \ 3 \times 3 & 3 \times 4 \end{array} \right] = \left[ \begin{array}{cc} 3 & 6 \ 9 & 12 \end{array} \right]. \)

This operation is fundamental when a matrix is used to scale transformations, and it often precedes more complex operations such as matrix addition or multiplication by another matrix.

Algebraic Methods

Algebraic methods in matrix operations involve using basic algebra principles to simplify or solve matrix equations. These methods can help you understand and systematically find solutions for even complex matrix operations.

• First, identify the operations needed and their order, such as multiplication before addition.


• Follow each operation step-by-step to maintain clarity and accuracy. For example, multiply a matrix by a scalar before adding it to another matrix, as we solved in the provided exercise.


• Using algebraic methods allows for precise hand calculations, ensuring a deeper understanding of how matrix operations work without over-relying on graphing utilities or calculators.

This step-by-step approach not only helps in getting correct answers but also builds a solid foundation for understanding higher-level mathematical concepts that utilize matrices.