Problem 16 - Algebra and Trigonometry

Question

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{rrr} {6} & {-3} & {5} \\ {6} & {0} & {-2} \\ {-4} & {2} & {-1} \end{array}\right], \quad B=\left[\begin{array}{rrr} {-3} & {5} & {1} \\ {-1} & {2} & {-6} \\ {2} & {0} & {4} \end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrices are:a) \(A+B = \left[\begin{array}{rrr} {3} & {2} & {6} \ {5} & {2} & {-8} \ {-2} & {2} & {3} \end{array}\right]\)b) \(A-B = \left[\begin{array}{rrr} {9} & {-8} & {4} \ {7} & {-2} & {4} \ {-6} & {2} & {-5} \end{array}\right]\)c) \(-4A = \left[\begin{array}{rrr} {-24} & {12} & {-20} \ {-24} & {0} & {8} \ {16} & {-8} & {4} \end{array}\right]\)d) \(3A+2B = \left[\begin{array}{rrr} {12} & {3} & {17} \ {16} & {6} & {-8} \ {-10} & {6} & {5} \end{array}\right]\)

Step 1: Addition of matrices

Add corresponding elements in matrix A and matrix B to find the matrix \(A + B\). This means, for example, for the first element, you would add 6 (from A) to -3 (from B). You do this for every single element, resulting in a new matrix.

Step 2: Subtraction of matrices

Subtract corresponding elements in matrix B from matrix A to find the matrix \(A - B\). This means, for example, for the first element, you would subtract -3 (from B) from 6 (from A). Again, do this for every single element to create a new matrix.

Step 3: Scalar multiplication

Multiply every element in matrix A by -4 to find the matrix \(-4A\). This means, for example, for the first element, you would multiply 6 (from A) by -4. Again, this operation is performed for every single element in the matrix.

Step 4: Linear combination

Multiply each element of matrix A by 3 and each element of matrix B by 2, then add the results, to find the matrix \(3A + 2B\). This operation is performed for every single element.

Key Concepts

Matrix Addition

When it comes to combining matrices, matrix addition is one of the most fundamental operations. Similar to how we add numbers, matrix addition involves combining two matrices of the same dimensions by adding their corresponding elements together. For example, consider two matrices, matrix A and matrix B:

• Matrix A = \(\begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \ \end{bmatrix}\)
• Matrix B = \(\begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \ \end{bmatrix}\)

The sum, denoted as \(A + B\), is obtained by taking each element from the same position in matrices A and B and adding them together to create a new matrix of the same dimensions:
\(A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \ \end{bmatrix}\)
This operation requires that both matrices have the same number of rows and columns; otherwise, addition is not defined. An important property of matrix addition is its commutativity, meaning that \(A + B = B + A\).

Matrix Subtraction

Just as you can add matrices, you can also subtract one matrix from another, provided they have the same dimensions. This process involves taking each corresponding pair of elements and performing a subtraction:

• Matrix A = \(\begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \ \end{bmatrix}\)
• Matrix B = \(\begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \ \end{bmatrix}\)

The result, \(A - B\), is formed as follows:
\(A - B = \begin{bmatrix} a_{11} - b_{11} & a_{12} - b_{12} \ a_{21} - b_{21} & a_{22} - b_{22} \ \end{bmatrix}\)
Matrix subtraction is not commutative, which means that \(A - B\) does not equal \(B - A\). Instead, subtracting B from A yields the additive inverse of subtracting A from B.

Scalar Multiplication of Matrices

In contrast to addition and subtraction, which involve two matrices, scalar multiplication involves a matrix and a single number (a scalar). To perform scalar multiplication, you simply multiply each element of the matrix by the scalar value. For example:

• Scalar, \(c\)
• Matrix A = \(\begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \ \end{bmatrix}\)

The product, \(cA\), would be:
\(cA = \begin{bmatrix} c \times a_{11} & c \times a_{12} \ c \times a_{21} & c \times a_{22} \ \end{bmatrix}\)
This operation stretches or shrinks the matrix by the scalar factor and is a fundamental operation in matrix algebra, enabling us to transform matrices in various ways.

Linear Combination of Matrices

A linear combination of matrices builds on both scalar multiplication and matrix addition. It is a new matrix created by adding together the scalar multiples of other matrices. It's like making a recipe where matrices are ingredients and scalars are the quantities.
For instance, if you have two matrices A and B, and two scalars c and d, the linear combination \(cA + dB\) would be calculated as follows:

• First, multiply each element of matrix A by scalar c.
• Then, multiply each element of matrix B by scalar d.
• Finally, add the two resulting matrices together to get the linear combination.

The ability to form linear combinations of matrices is crucial as it underpins various concepts in linear algebra, including matrix transformations, systems of linear equations, and more.