Problem 4 - Precalculus: Functions and Graphs

Question

Find, if possible, \(A+B, A-B, 2 A,\) and \(-3 B\). $$A=\left[\begin{array}{rrr} 0 & -2 & 7 \\ 5 & 4 & -3 \end{array}\right], \quad B=\left[\begin{array}{lll} 8 & 4 & 0 \\ 0 & 1 & 4 \end{array}\right]$$

Step-by-Step Solution

Answer

\(A+B = \left[\begin{smallmatrix} 8 & 2 & 7 \\ 5 & 5 & 1 \end{smallmatrix}\right], A-B = \left[\begin{smallmatrix} -8 & -6 & 7 \\ 5 & 3 & -7 \end{smallmatrix}\right], 2A = \left[\begin{smallmatrix} 0 & -4 & 14 \\ 10 & 8 & -6 \end{smallmatrix}\right], -3B = \left[\begin{smallmatrix} -24 & -12 & 0 \\ 0 & -3 & -12 \end{smallmatrix}\right]."

Step 1: Check matrix dimensions for addition and subtraction

In order to add or subtract two matrices, they must have the same dimensions. The matrix \(A\) is a \(2 \times 3\) matrix and the matrix \(B\) is also a \(2 \times 3\) matrix. Since both matrices have the same dimensions, we can proceed with addition and subtraction.

Step 2: Calculate \(A + B\)

To find \(A + B\), add each corresponding element from matrices \(A\) and \(B\). \[A + B = \left[\begin{array}{rrr} 0+8 & -2+4 & 7+0 \ 5+0 & 4+1 & -3+4 \end{array} \right] = \left[\begin{array}{ccc} 8 & 2 & 7 \ 5 & 5 & 1 \end{array} \right]\]

Step 3: Calculate \(A - B\)

To find \(A - B\), subtract each corresponding element of matrix \(B\) from matrix \(A\). \[A - B = \left[\begin{array}{rrr} 0-8 & -2-4 & 7-0 \ 5-0 & 4-1 & -3-4 \end{array} \right] = \left[\begin{array}{ccc} -8 & -6 & 7 \ 5 & 3 & -7 \end{array} \right]\]

Step 4: Calculate \(2A\)

To find \(2A\), multiply each element of matrix \(A\) by 2. \[2A = \left[\begin{array}{rrr} 2\cdot0 & 2\cdot(-2) & 2\cdot7 \ 2\cdot5 & 2\cdot4 & 2\cdot(-3) \end{array} \right] = \left[\begin{array}{ccc} 0 & -4 & 14 \ 10 & 8 & -6 \end{array} \right]\]

Step 5: Calculate \(-3B\)

To find \(-3B\), multiply each element of matrix \(B\) by \(-3\). \[-3B = \left[\begin{array}{rrr} -3\cdot8 & -3\cdot4 & -3\cdot0 \ -3\cdot0 & -3\cdot1 & -3\cdot4 \end{array} \right] = \left[\begin{array}{ccc} -24 & -12 & 0 \ 0 & -3 & -12 \end{array} \right]\]

Key Concepts

Matrix Addition

Matrix addition is a straightforward process, as long as you remember the golden rule: matrices must have the same dimensions to be added together. If you're working with matrices of different sizes, the addition cannot be performed. For our matrices, both have dimensions of 2 rows and 3 columns, denoted as a 2x3 dimension.

• To add matrix A to matrix B (denoted as A + B), each element in matrix A is added to the corresponding element in matrix B.
• The elements must line up perfectly: the first row of A with the first row of B, and so on.

In our example, by adding each element from matrix A to the corresponding one in matrix B, we computed:\[A + B = \begin{bmatrix} 8 & 2 & 7 \ 5 & 5 & 1 \end{bmatrix}\]This results in a new matrix of the same dimensions, where each element is a sum of the elements from A and B.

Matrix Subtraction

Just like matrix addition, matrix subtraction also requires that the matrices involved have the same dimensions. If you obey this rule, you will be able to smoothly execute the subtraction process.

• Matrix subtraction is performed by taking each element from matrix A and subtracting the corresponding element from matrix B (denoted as A - B).
• Make sure that each corresponding element aligns as they did in addition: row to row, column to column.

Once again using our sample matrices A and B, each entry of B was subtracted from A:\[A - B = \begin{bmatrix} -8 & -6 & 7 \ 5 & 3 & -7 \end{bmatrix}\]This result is a new matrix, calculated by subtracting B's elements from A's respective elements.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a single number, known as a scalar.

• This operation scales the matrix but does not change its dimensions.
• If matrix A is multiplied by scalar 2, every element of A will be doubled.

Using matrix A, when performing scalar multiplication with 2, our result is:\[2A = \begin{bmatrix} 0 & -4 & 14 \ 10 & 8 & -6 \end{bmatrix}\]In another instance, when multiplying matrix B by -3, each of B's elements is multiplied by -3, yielding:\[-3B = \begin{bmatrix} -24 & -12 & 0 \ 0 & -3 & -12 \end{bmatrix}\]Scalar multiplication is an efficient way to uniformly scale a matrix.

Matrix Dimensions

Matrix dimensions are denoted by the number of rows and columns a matrix contains, written as rows \(\times\) columns. Understanding these dimensions is key to performing matrix operations like addition and subtraction because both require matrices to share identical dimensions.

• For instance, matrix A in our task is a 2x3 matrix, simply meaning it has 2 rows and 3 columns.
• If another matrix is not 2x3, it cannot be added to or subtracted from our matrix A directly.

Having the correct dimensions ensures that operations like component-wise addition and subtraction are valid, allowing you to accurately compute results without errors.