Problem 22 - Algebra and Trigonometry

Question

The matrices \(A, B, C, D, E, F, G\) and \(H\) are defined as follows. $$ A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right] $$ $$ D=\left[\begin{array}{ll}{7} & {3}\end{array}\right] \quad E=\left[\begin{array}{l}{1} \\ {2} \\ {0}\end{array}\right] \quad F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] $$ $$ G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right] \quad H=\left[\begin{array}{rr}{3} & {1} \\ {2} & {-1}\end{array}\right] $$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$ \begin{array}{ll}{\text { (a) } C-B} & {\text { (b) } 2 C-6 B}\end{array} $$

Step-by-Step Solution

Answer

(a) \(C-B=\begin{bmatrix} -1 & -3 & -5 \\ -1 & 3 & -6 \end{bmatrix}\), (b) \(2C-6B=\begin{bmatrix} -14 & -8 & -30 \\ -6 & 10 & -24 \end{bmatrix}\).

Step 1: Determine Compatibility for Subtraction (C-B)

To subtract two matrices, they must have the same dimensions. Matrix \(C\) is a \(2 \times 3\) matrix, and matrix \(B\) is also a \(2 \times 3\) matrix. Since they are the same size, the operation \(C - B\) can be performed.

Step 2: Subtract Matrices (C-B)

Subtract corresponding elements of matrix \(B\) from matrix \(C\): \[C - B = \left[\begin{array}{rrr}2 & -\frac{5}{2} & 0 \ 0 & 2 & -3\end{array}\right] - \left[\begin{array}{rrr}3 & \frac{1}{2} & 5 \ 1 & -1 & 3\end{array}\right]\]\[= \left[\begin{array}{rrr} 2-3 & -\frac{5}{2}-\frac{1}{2} & 0-5 \ 0-1 & 2+1 & -3-3 \end{array}\right]\]\[= \left[\begin{array}{rrr} -1 & -3 & -5 \ -1 & 3 & -6 \end{array}\right]\]

Step 3: Determine Compatibility for Scalar Multiplication (2C and 6B)

Scalar multiplication involves multiplying each element of a matrix by the scalar. Both matrix \(C\) and matrix \(B\) can be multiplied by scalars as they are \(2 \times 3\) matrices.

Step 4: Perform Scalar Multiplication (2C and 6B)

Compute the products for \(2C\) and \(6B\):\[2C = 2 \times \left[\begin{array}{rrr}2 & -\frac{5}{2} & 0 \ 0 & 2 & -3\end{array}\right] \]\[= \left[\begin{array}{rrr} 4 & -5 & 0 \ 0 & 4 & -6 \end{array}\right]\]And, \[6B = 6 \times \left[\begin{array}{rrr}3 & \frac{1}{2} & 5 \ 1 & -1 & 3\end{array}\right] \]\[= \left[\begin{array}{rrr} 18 & 3 & 30 \ 6 & -6 & 18 \end{array}\right]\]

Step 5: Subtract Scalar Multiplications (2C - 6B)

Subtract the results of \(6B\) from \(2C\):\[2C - 6B = \left[\begin{array}{rrr}4 & -5 & 0 \ 0 & 4 & -6\end{array}\right] - \left[\begin{array}{rrr}18 & 3 & 30 \ 6 & -6 & 18\end{array}\right]\]\[= \left[\begin{array}{rrr} 4-18 & -5-3 & 0-30 \ 0-6 & 4+6 & -6-18 \end{array}\right]\]\[= \left[\begin{array}{rrr} -14 & -8 & -30 \ -6 & 10 & -24 \end{array}\right]\]

Key Concepts

Matrix Subtraction

Matrix subtraction is the process of subtracting corresponding elements of one matrix from another. For this to happen, both matrices must have identical dimensions. This means that they must have the same number of rows and columns. If this condition is not met, matrix subtraction is not possible.
When subtracting matrices, each element in the resulting matrix is calculated by subtracting the element in one matrix from the element in the same position of the other matrix. As an example, if you have matrices A and B of dimensions 2x3, then the matrix subtraction A - B is performed by subtracting the element in the first row and first column of B from the element in the first row and first column of A, continuing this process for each corresponding element throughout the matrices.

• Matrix Subtraction Example: If A = \( \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \end{bmatrix} \) and B = \( \begin{bmatrix} b_{11} & b_{12} & b_{13} \ b_{21} & b_{22} & b_{23} \end{bmatrix} \), A - B would be \( \begin{bmatrix} a_{11} - b_{11} & a_{12} - b_{12} & a_{13} - b_{13} \ a_{21} - b_{21} & a_{22} - b_{22} & a_{23} - b_{23} \end{bmatrix} \).
• The operation is carried out element by element.

To summarize, matrix subtraction is all about making sure the matrices are of the same size and then performing straightforward element-wise subtraction.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a scalar, which is simply a constant number. This process is quite different from matrix multiplication, which involves rows and columns interacting.
When performing scalar multiplication, the number you multiply with is called the scalar. You apply this scalar to each element of the matrix individually. This does not change the dimensions of the matrix—it will still be the same size—but it alters the values of each element according to the multiplication.

• For instance, if you multiply a 2x3 matrix A by a scalar k, such as 2, each element \( a_{ij} \) becomes \( 2 \times a_{ij} \).
• Scalar multiplication does not affect the structure (number of rows and columns) of the matrix.

Understanding scalar multiplication is foundational when you need to perform operations that involve scaling the values within your matrices. It's crucial to remember that you simply multiply each number in the matrix by the scalar, impacting each element independently.

Matrix Dimensions

The dimensions of a matrix are fundamental to understanding how you can use or manipulate it within mathematical operations. Matrix dimensions are expressed in terms of the number of rows and columns a matrix contains, typically presented in the format "rows \( \times \) columns."
For a matrix to be used in operations like addition or subtraction, it must have the same dimensions as the matrix it is being paired with. Whether a matrix is conformable for these operations is directly determined by its dimensions.

• A 2x3 matrix has 2 rows and 3 columns.
• If you want to add or subtract matrices, make sure they have the same dimensions.

Knowing the dimensions is also crucial when it comes to matrix multiplication or even when setting up matrices for linear equations. It tells you the structure and how matrices can potentially interact with other matrices or vectors.

Element-wise Operations

Element-wise operations are basic yet powerful actions performed on matrices that involve applying a certain operation to each individual component or element of a matrix. Matrix addition, subtraction, and scalar multiplication are all examples where element-wise operations are used.
To perform an element-wise operation, look at the operation to be done (addition, subtraction, etc.) and apply it specifically and independently to each corresponding element from the involved matrices or matrix and scalar.

• Think of adding matrices like adding two spreadsheets of numbers cell-by-cell.
• Subtractions work the same way but instead take away value from each corresponding element.
• Scalar multiplication also uses this method but involves a single matrix and multiplies each element by the same scalar.

This approach allows you to manipulate entire datasets with clear, straightforward calculations, ensuring each element is treated as part of a larger whole without losing its individual identity in the calculation process.