Problem 34 - Algebra 2

Question

Use matrices \(A, B, C,\) and \(D\) to find the following. $$A=\left[\begin{array}{rr}{5} & {7} \\ {-1} & {6} \\ {3} & {-9}\end{array}\right] \quad B=\left[\begin{array}{ll}{8} & {3} \\ {5} & {1} \\\ {4} & {4}\end{array}\right] \quad C=\left[\begin{array}{rr}{0} & {4} \\\ {-2} & {5} \\ {7} & {-1}\end{array}\right] \quad D=\left[\begin{array}{ll}{6} & {2} \\ {9} & {0} \\ {-3} & {0}\end{array}\right]$$ \(C+\frac{1}{3} D\)

Step-by-Step Solution

Answer

The matrix is \( \begin{bmatrix} 2 & \frac{14}{3} \\ 1 & 5 \\ 6 & -1 \end{bmatrix} \).

Step 1: Simplify \( \frac{1}{3} D \)

To find \( \frac{1}{3} D \), multiply each element of matrix \( D \) by \( \frac{1}{3} \).\[D = \begin{bmatrix} 6 & 2 \ 9 & 0 \ -3 & 0 \end{bmatrix}\]Therefore, \( \frac{1}{3} D \) is:\[\frac{1}{3} D = \begin{bmatrix} \frac{1}{3} \cdot 6 & \frac{1}{3} \cdot 2 \ \frac{1}{3} \cdot 9 & \frac{1}{3} \cdot 0 \ \frac{1}{3} \cdot (-3) & \frac{1}{3} \cdot 0 \end{bmatrix} = \begin{bmatrix} 2 & \frac{2}{3} \ 3 & 0 \ -1 & 0 \end{bmatrix}\]

Step 2: Add Matrices \( C \) and \( \frac{1}{3} D \)

Add each corresponding element of matrices \( C \) and \( \frac{1}{3} D \).\[C = \begin{bmatrix} 0 & 4 \ -2 & 5 \ 7 & -1 \end{bmatrix}\]Adding \( C \) and \( \frac{1}{3} D \):\[C + \frac{1}{3} D = \begin{bmatrix} 0 + 2 & 4 + \frac{2}{3} \ -2 + 3 & 5 + 0 \ 7 + (-1) & -1 + 0 \end{bmatrix} = \begin{bmatrix} 2 & \frac{14}{3} \ 1 & 5 \ 6 & -1 \end{bmatrix}\]

Step 3: Conclusion

The resulting matrix from \( C + \frac{1}{3} D \) is \[\begin{bmatrix} 2 & \frac{14}{3} \ 1 & 5 \ 6 & -1 \end{bmatrix}.\]

Key Concepts

Matrix Scalar Multiplication

Matrix scalar multiplication is a fundamental operation within matrix algebra. It involves multiplying each entry of a given matrix by a constant, called a scalar. This operation is straightforward but essential for manipulating matrices for various applications, such as physics, computer graphics, and economics.

For example, in the exercise, we have matrix \(D\) given as \[D = \begin{bmatrix} 6 & 2 \ 9 & 0 \ -3 & 0 \end{bmatrix}\] To perform matrix scalar multiplication by \(\frac{1}{3}\), simply multiply each entry of matrix \(D\) by \(\frac{1}{3}\). This results in a new matrix:\[\frac{1}{3} D = \begin{bmatrix} \frac{6}{3} & \frac{2}{3} \ \frac{9}{3} & 0 \ \frac{-3}{3} & 0 \end{bmatrix} = \begin{bmatrix} 2 & \frac{2}{3} \ 3 & 0 \ -1 & 0 \end{bmatrix}\]

Every element of the original matrix is scaled by \(\frac{1}{3}\), which means each entry must be multiplied by this value. The outcome is a matrix with the same dimensions as the original, showcasing how scalar multiplication only affects the magnitudes of the entries, not the overall shape of the matrix.

Matrix Operations

Matrix operations encompass several fundamental procedures that allow us to manipulate and solve matrices effectively. The two basic operations are matrix addition and matrix multiplication.

In our exercise, we focus on matrix addition. This involves adding matrices by summing their corresponding elements. It is crucial for this operation that matrices involved must have the same dimensions.

For instance, consider matrices \(C\) and \(\frac{1}{3} D\) as described in the exercise: \[C = \begin{bmatrix} 0 & 4 \ -2 & 5 \ 7 & -1 \end{bmatrix}\] \[\frac{1}{3} D = \begin{bmatrix} 2 & \frac{2}{3} \ 3 & 0 \ -1 & 0 \end{bmatrix}\]

To add them, correspond each element:- For the first row: \(0 + 2\) and \(4 + \frac{2}{3}\)- For the second row: \(-2 + 3\) and \(5 + 0\)- For the third row: \(7 + (-1)\) and \(-1 + 0\)The result of this operation will be \[C + \frac{1}{3} D = \begin{bmatrix} 2 & \frac{14}{3} \ 1 & 5 \ 6 & -1 \end{bmatrix}\]Each entry is computed by addition, highlighting the simplicity and elegance of combining different matrices.

Matrices

Matrices are a mathematical representation used ubiquitously in a variety of fields to solve problems related to linear equations, transformations, and other algebraic equations. A matrix is essentially a rectangular array of numbers arranged in rows and columns. These arrays can represent complex systems beautifully and concisely.

When working with matrices, it’s essential to understand their dimensions, which are given by the number of rows and columns. For example, a matrix with three rows and two columns is described as a \(3 \times 2\) matrix. Identifying dimensions is a necessary step before performing operations such as addition or multiplication, as only matrices of certain compatible dimensions can be combined in specific ways.

Matrices \(A, B, C,\) and \(D\) in the given exercise each demonstrate an understanding of structure and component manipulation. Whether it's multiplying a matrix by a scalar or adding two matrices, these operations serve to modify or transform the data captured by these matrices. In doing so, they provide an efficient means to solve and analyze mathematical problems.