Problem 25 - 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. $$ \begin{array}{ll}{\text { (a) } A D} & {\text { (b) } D A}\end{array} $$

Step-by-Step Solution

Answer

AD is not possible. DA is \([14, -14]\).

Step 1: Determine the Dimensions for AD

Matrix A is a 2x2 matrix and matrix D is a 1x2 matrix. To multiply two matrices (in order AD), the number of columns in the first matrix must match the number of rows in the second matrix.

Step 2: Check Compatibility for AD

Matrix A has 2 columns, while matrix D has 1 row. Since the number of columns in A (2) does not equal the number of rows in D (1), matrix multiplication AD is not possible.

Step 3: Determine the Dimensions for DA

Matrix D is a 1x2 matrix and matrix A is a 2x2 matrix. To multiply two matrices (in order DA), the number of columns in the first matrix must match the number of rows in the second matrix.

Step 4: Check Compatibility for DA

Matrix D has 2 columns, which matches the 2 rows in Matrix A. Therefore, the multiplication DA is possible and will result in a 1x2 matrix.

Step 5: Perform the Multiplication DA

To compute DA: \[ D = \begin{bmatrix} 7 & 3 \end{bmatrix}, \ A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \] Multiply the row from D by each column in A: - For the first column of A: \((7 * 2) + (3 * 0) = 14 + 0 = 14\).- For the second column of A: \((7 * -5) + (3 * 7) = -35 + 21 = -14\).Therefore, \( DA = \begin{bmatrix} 14 & -14 \end{bmatrix} \).

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial when working with matrices, especially in matrix operations like multiplication. A matrix is defined by the number of its rows and columns. This is often expressed as "m x n", where "m" is the number of rows and "n" is the number of columns. For example, matrix A, represented as \( \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \), is a 2x2 matrix. This means it has 2 rows and 2 columns.
Different matrices in a given problem may have diverse dimensions, such as matrix D, which is a 1x2 matrix because it has 1 row and 2 columns. Recognizing these dimensions immediately can guide whether certain operations, like multiplication, can be performed. Always check the dimensions first to know the structure and size of the matrices involved.

Matrix Compatibility

Matrix compatibility in multiplication means that the number of columns in the first matrix must equal the number of rows in the second matrix. This rule determines whether two matrices can be multiplied together. If we take the matrices in the exercise, for instance, matrix A is a 2x2 matrix, while matrix D is a 1x2 matrix.

• For the operation \( AD \), we check if the number of columns in A (2) is the same as the number of rows in D (1). Since they do not match, the multiplication is not possible.
• However, for the operation \( DA \), D has 2 columns and A has 2 rows, making them compatible for multiplication.

Not all matrices can be multiplied, even if they have the same total number of elements. Always verify the inner dimensions (the number of columns in the first matrix with the number of rows in the second) to ensure they align for successful multiplication.

Matrix Algebra

Matrix algebra involves various operations such as addition, subtraction, and multiplication. Among these, multiplication is particularly important and is based on specific rules, such as those regarding compatibility. When multiplying two matrices, the resulting matrix will have dimensions based on the rows of the first matrix and the columns of the second matrix. For instance, when you multiply matrix D \( \begin{bmatrix} 7 & 3 \end{bmatrix} \) with matrix A \( \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \), you perform a series of dot products. For the first entry of the resulting matrix DA, you multiply and sum the elements of D’s row with the corresponding elements from A’s first column: \[(7 \times 2) + (3 \times 0) = 14 + 0 = 14\]
For the second entry, you use the second column of A: \[(7 \times -5) + (3 \times 7) = -35 + 21 = -14\]This results in \( DA = \begin{bmatrix} 14 & -14 \end{bmatrix} \). Each product like this can be broken down into simpler parts to ease understanding. Practicing with examples simplifies complex algebraic concepts, reinforcing how mathematical operations affect dimensions and compatibility.