Problem 23 - 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. $$ \begin{array}{ll}{\text { (a) } 5 A} & {\text { (b) } C-5 A}\end{array} $$

Step-by-Step Solution

Answer

(a) \(5A = \begin{bmatrix} 10 & -25 \\ 0 & 35 \end{bmatrix}\). (b) Cannot perform \(C - 5A\) due to dimension mismatch.

Step 1: Scaling Matrix A by 5

To calculate \(5A\), multiply each element of matrix \(A\) by the scalar 5.Given:\[A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\]Calculation:\[5A = 5 \times \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} = \begin{bmatrix} 5 \times 2 & 5 \times (-5) \ 5 \times 0 & 5 \times 7 \end{bmatrix} = \begin{bmatrix} 10 & -25 \ 0 & 35 \end{bmatrix}\]

Step 2: Check Dimensions for Subtraction

Before performing the operation \(C - 5A\), ensure both matrices \(C\) and \(5A\) have the same dimensions. Matrix \(C\) is a \(2 \times 3\) matrix:\[C = \begin{bmatrix} 2 & -\frac{5}{2} & 0 \ 0 & 2 & -3 \end{bmatrix} \]Matrix \(5A\) is a \(2 \times 2\) matrix:\[5A = \begin{bmatrix} 10 & -25 \ 0 & 35 \end{bmatrix}\]Since one is \(2 \times 3\) and the other is \(2 \times 2\), they did not have the same dimensions.

Step 3: Conclude Reasoning for Operation (b)

Since matrices \(C\) and \(5A\) do not have the same dimensions (the first has 3 columns and the second has 2 columns), it is not possible to subtract them. Matrix subtraction requires both matrices to be of exactly the same size.

Key Concepts

Matrix Multiplication

Matrix multiplication is a binary operation that produces a matrix from two matrices. To multiply two matrices, we must first check their dimensions. **Matrix A** with dimensions of \(m \times n\) can only be multiplied by **Matrix B** of dimensions \(n \times p\). This requirement ensures that the number of columns in the first matrix matches the number of rows in the second matrix.

To perform the multiplication, take each element of a row from the first matrix and a column from the second matrix, multiply them, and then sum the results to get an element of the resulting matrix. This process has to be repeated for each row of the first matrix and each column of the second matrix to populate the entire resulting matrix. The resulting matrix will have dimensions \(m \times p\).

• Ensure compatible dimensions \((m \times n) \times (n \times p)\)
• Multiply and sum the corresponding rows and columns
• Resulting matrix is \(m \times p\)

Matrix multiplication is not commutative, meaning that \(AB eq BA\) in most cases. Therefore, always pay close attention to the order when performing matrix multiplication.

Matrix Subtraction

Matrix subtraction is straightforward, but there is a critical requirement that both matrices must have the same dimensions. For instance, if Matrix C is a \(2 \times 3\) matrix, then Matrix D must also be \(2 \times 3\) to perform subtraction.

To subtract one matrix from another:

• Ensure both matrices are of the same size
• Subtract corresponding elements

Each element in the resulting matrix is found by directly subtracting corresponding elements from the two matrices. If the matrices are not compatible in size, subtraction cannot be performed, as seen in the problem with matrices **C** and **5A**, which have different dimensions \((2 \times 3)\) and \((2 \times 2)\), respectively.

Matrix Dimensions

Understanding matrix dimensions is crucial to performing any matrix operations, including multiplication, addition, and subtraction. The dimensions of a matrix are represented as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns.

Identifying the dimensions aids in determining:

• Compatibility for operations like addition, subtraction, and multiplication
• Size of the resulting matrix post multiplication

Reading matrix dimensions is simple: count the rows and then count the columns. This knowledge ensures appropriate operations are selected and performed correctly, as some operations require stringent size conditions. For instance, matrix addition and subtraction require identical dimensions, while multiplication requires specific row-column alignment.

Scalar Multiplication

Scalar multiplication involves a single number, a scalar, and a matrix. This operation is simpler than matrix multiplication. Here, you multiply each entry of the matrix by the scalar.

For example, in the exercise, matrix **A** is multiplied by a scalar 5. This step involves multiplying every element of matrix **A** by 5:

• Multiply every element in the matrix by the scalar
• Retain the dimensions of the original matrix

The process can be represented mathematically as:

• Original matrix: \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\)
• Scalar multiplication with 5 gives: \(5A = \begin{bmatrix} 5a & 5b \ 5c & 5d \end{bmatrix}\)

Scalar multiplication does not alter the shape of the matrix and is a crucial step in many more complex matrix operations and algebraic manipulations.