Problem 37 - Algebra and Trigonometry

Question

Perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} {4} & {0} \\ {-3} & {5} \\ {0} & {1} \end{array}\right] \quad B=\left[\begin{array}{rr} {5} & {1} \\ {-2} & {-2} \end{array}\right] \quad C=\left[\begin{array}{rr} {1} & {-1} \\ {-1} & {1} \end{array}\right] $$ $$ 4 B-3 C $$

Step-by-Step Solution

Answer

The result of the matrix operation \(4B - 3C\) is \(\left[\begin{array}{rr} {17} & {7} \ {-5} & {-11} \end{array}\right].\)

Step 1: Multiplication by Scalars

Multiply matrix \(B\) by 4 and matrix \(C\) by 3. This is done by multiplying every entry of the matrix by the scalar. The result is \[4B = \left[\begin{array}{rr} {20} & {4} \ {-8} & {-8} \end{array}\right], 3C = \left[\begin{array}{rr} {3} & {-3} \ {-3} & {3} \end{array}\right]\]

Step 2: Matrix Subtraction

Subtract the two resulting matrices from Step 1. Subtract corresponding elements from each matrix: \(4B - 3C = \left[\begin{array}{rr} {20-3} & {4-(-3)} \ {-8-(-3)} & {-8-3} \end{array}\right].\)

Step 3: Final Calculation

Simplify the computations in each matrix entry. The resulting matrix is the answer to the operation \(4B - 3C = \left[\begin{array}{rr} {17} & {7} \ {-5} & {-11} \end{array}\right]\).

Key Concepts

Matrix Multiplication

Matrix multiplication involves a special way of combining two matrices to produce a new matrix. Unlike multiplying numbers, matrix multiplication is not commutative, meaning that the order of matrices matters. To multiply a matrix by another matrix, you follow these steps:

• Ensure that the number of columns in the first matrix matches the number of rows in the second matrix. If not, the multiplication is not defined.
• Multiply each element of the rows of the first matrix by the corresponding element of the columns of the second matrix and sum these products to find the resulting matrix element.

For example, if you multiply an element in the first row and first column of the first matrix by the corresponding elements in the first column of the second matrix, you get an element for the first row and first column of the resulting matrix.
This process continues for each row and column until all elements are calculated.
Matrix multiplication is used extensively in mathematics for transforming data or solving systems of linear equations. It is important to remember that even if two matrices can be multiplied, the resulting matrix will have different dimensions from the original matrices.

Matrix Subtraction

Matrix subtraction is the process of subtracting corresponding elements from two matrices of the same size. For matrix subtraction to be valid, both matrices must have exactly the same dimensions.
To perform matrix subtraction, follow these simple steps:

• Ensure both matrices involved have the same number of rows and columns.
• For each element in the matrices, subtract the corresponding element in the second matrix from the element in the first matrix.

For example, to subtract two matrices, subtract the top left element of the second matrix from the top left element of the first matrix. Continue this for each position until all elements have been processed.
Matrix subtraction is often used in solving equations involving matrices or in adjusting matrices through specific transformations. The process is straightforward, but the key is ensuring both matrices are compatible in size before performing the operation.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a scalar (a constant number). This operation scales the size of the matrix’s elements uniformly and is quite straightforward.
To multiply a matrix by a scalar, simply:

• Take each element in the original matrix.
• Multiply it by the scalar value.
• Replace each original element with the result of the multiplication.

For example, if you have a matrix \(B\) and you want to multiply it by 4, you multiply each entry in \(B\) by 4 to get the new matrix \(4B\).
This operation does not change the structure of the matrix itself, just the values of the elements. By using scalar multiplication, you can scale and transform matrices according to the problem's needs. This operation is particularly useful in various fields like physics and computer graphics where scaling of vectors and transformations are needed.