Problem 24 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The product \( DA \) is \( \left[\begin{array}{rr} 14 & -14 \end{array} \right] \).

Step 1: Determine Matrix Dimensions

First, note the dimensions of matrices involved. Matrix \( D \) is a 1x2 matrix (1 row and 2 columns), and matrix \( A \) is a 2x2 matrix (2 rows and 2 columns).

Step 2: Check Compatibility for Multiplication

For matrix multiplication to be possible, the number of columns in the first matrix (\( D \)) must equal the number of rows in the second matrix (\( A \)). Since \( D \) has 2 columns and \( A \) has 2 rows, they are compatible for multiplication.

Step 3: Perform Matrix Multiplication

Calculate the product \( DA \) by multiplying each element of the row of matrix \( D \) with the corresponding element of the column of matrix \( A \), and summing up the products for each position.The product \( DA \) is a 1x2 matrix:\[DA = \left[\begin{array}{rr} 7 & 3 \end{array}\right] \left[\begin{array}{rr} 2 & -5 \ 0 & 7 \end{array} \right] = \left[\begin{array}{rr} (7 \times 2) + (3 \times 0) & (7 \times -5) + (3 \times 7) \end{array} \right]\]Thus,\[DA = \left[\begin{array}{rr} 14 & -35 + 21 \end{array} \right]\]Finally,\[DA = \left[\begin{array}{rr} 14 & -14 \end{array} \right]\]

Step 4: Final Result

The resulting matrix from \( DA \) multiplication operation is \( \left[\begin{array}{rr} 14 & -14 \end{array} \right] \).

Key Concepts

Matrix Dimensions

When dealing with matrices, understanding their dimensions is crucial. Matrix dimensions are expressed as "rows x columns."
For example, if a matrix has 3 rows and 2 columns, its dimensions would be "3x2."
Each entry is located by its row and column position. For instance, the top-left entry of a matrix is the first row, first column. In the given exercise, matrix \( D \) has dimensions of 1x2, meaning it consists of one row and two columns. Meanwhile, matrix \( A \) is a 2x2 matrix, having two rows and two columns.
Understanding dimensions helps in visualizing the matrix's shape and is the first step in determining if operations like multiplication are possible.

Matrix Compatibility

Matrix compatibility is vital when performing operations like multiplication. A common stumbling block is knowing when two matrices can be multiplied.
The key rule is that the number of columns in the first matrix must match the number of rows in the second matrix.
This ensures each element in a row from the first matrix corresponds to each element in a column from the second matrix during multiplication. For the matrices in the exercise, matrix \( D \) is a 1x2 matrix, and matrix \( A \) is a 2x2 matrix.
Here, the number of columns in \( D \) (which is 2) matches the number of rows in \( A \) (also 2).
This means they are compatible for multiplication, and the resulting product will have dimensions based on the remaining dimensions—here, 1x2.

Matrix Operations

Matrix operations can seem daunting at first, but with practice, they become more intuitive. Let's focus on matrix multiplication, a fundamental operation in linear algebra.
In matrix multiplication, an entry in the product matrix is found by multiplying corresponding elements from the rows of the first matrix and columns of the second matrix and summing them up. When you multiply \( D \) and \( A \) from the exercise:

• First, take the elements in the sole row of \( D \) (7 and 3) and multiply them by their counterparts in each column of \( A \).
• The first element in the resulting matrix is computed by \((7 \times 2) + (3 \times 0) = 14\).
• The second element is computed by \((7 \times -5) + (3 \times 7) = -35 + 21 = -14\).

Therefore, the resulting matrix \( DA \) equals \( \left[\begin{array}{rr} 14 & -14 \end{array} \right] \).
Understanding these steps ensures a solid grasp of matrix multiplication fundamentals.