Problem 21 - 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. $$ \begin{array}{ll}{\text { (a) } B+C} & {\text { (b) } B+F}\end{array} $$

Step-by-Step Solution

Answer

(a) \( B + C = \begin{bmatrix} 5 & -2 & 5 \\ 1 & 1 & 0 \end{bmatrix} \); (b) \( B + F \) cannot be performed due to different dimensions.

Step 1: Check Dimensions for Addition B + C

Matrix addition requires both matrices to have the same dimensions. Matrix \( B \) has dimensions 2x3 and matrix \( C \) also has dimensions 2x3.

Step 2: Perform Addition for B + C

Since matrices \( B \) and \( C \) have the same dimensions, we can add them element-wise. Calculate the sum: \( B+C = \begin{bmatrix} 3+2 & \frac{1}{2} + (-\frac{5}{2}) & 5 + 0 \ 1+0 & -1 + 2 & 3 + (-3) \end{bmatrix} = \begin{bmatrix} 5 & -2 & 5 \ 1 & 1 & 0 \end{bmatrix} \)

Step 3: Check Dimensions for Addition B + F

Matrix \( B \) has dimensions 2x3, while matrix \( F \) has dimensions 3x3. In order to add two matrices, they must have the same dimensions.

Step 4: Conclusion for B + F

Since matrices \( B \) and \( F \) do not have the same dimensions, the addition operation \( B + F \) cannot be performed.

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in matrix algebra. It involves combining two matrices by adding their corresponding elements. To perform matrix addition, both matrices involved must have the exact same dimensions.

For instance, if you have two matrices, Matrix A and Matrix B, both with dimensions 2x3, you can perform matrix addition by simply adding each corresponding element from Matrix A to Matrix B. This operation is known as element-wise addition.

Thus, if \[A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}\] \[B = \begin{bmatrix} 7 & 8 & 9 \ 10 & 11 & 12 \end{bmatrix}\], the result of \( A + B \) is \[\begin{bmatrix} 8 & 10 & 12 \ 14 & 16 & 18 \end{bmatrix}\].

Matrix addition is commutative, meaning \( A + B \) is the same as \( B + A \), and associative, which means \( (A + B) + C = A + (B + C) \). Therefore, always remember to check the dimensions first before adding matrices.

Matrix Dimensions

Understanding matrix dimensions is crucial in performing any matrix operation. The dimension of a matrix is defined by the number of rows and columns it contains. It is usually denoted as m x n, where m represents the number of rows, and n represents the number of columns.

For example, a 2x3 matrix has 2 rows and 3 columns, while a 3x2 matrix has 3 rows and 2 columns. The compatibility of matrices for addition strictly depends on these dimensions. If two matrices do not share the same dimensions, they cannot be added together.

When dealing with matrices:

• Ensure that both matrices have the same number of rows and columns for addition.
• For multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix.
• Sizes of matrices can be determined simply by counting their rows and columns.

These rules ensure that matrix operations are conducted correctly and consistently.

Element-wise Addition

Element-wise addition is the process of adding two matrices by individually summing their corresponding elements. This operation is only possible if the two matrices have identical dimensions.

To visualize element-wise addition, consider two matrices:

• Matrix X: \\[\begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}\]
• Matrix Y: \\[\begin{bmatrix} 1 & 3 \ 5 & 7 \end{bmatrix}\]

Adding them involves calculating: \[ X+Y=\begin{bmatrix} 2+1 & 4+3 \ 6+5 & 8+7 \end{bmatrix}=\begin{bmatrix} 3 & 7 \ 11 & 15 \end{bmatrix}\]

This element-wise addition shows how each position in the resulting matrix is computed separately by summing elements from the same position in both matrices. This way, each matrix retains its structural integrity while combining values.

Matrix Algebra

Matrix algebra encompasses a set of operations that can be performed on matrices, including addition, subtraction, multiplication, and more advanced operations like inversion. The rules of matrix algebra are essential for solving complex mathematical and engineering problems.

Some basic points to remember in matrix algebra:

• Matrix addition and subtraction require matrices to have the same dimensions.
• Matrix multiplication has different rules: the number of columns of the first matrix must match the number of rows of the second.
• The identity matrix is a special kind of square matrix that, when multiplied by any matrix, doesn’t change that matrix’s value.
• Matrices can represent systems of equations, allowing for efficient solving using matrix manipulation techniques.

These operations form the building blocks of matrix algebra, enabling practitioners to handle a wide array of applications efficiently and accurately.