Problem 26 - 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. $$ \begin{array}{ll}{\text { (a) } D H} & {\text { (b) } H D}\end{array} $$

Step-by-Step Solution

Answer

(a) DH = \(\begin{bmatrix} 27 & 4 \end{bmatrix}\); (b) HD is undefined.

Step 1: Check Matrix Multiplication Validity for DH

To determine if matrix multiplication is possible, recall that the number of columns in the first matrix must equal the number of rows in the second matrix.Matrix \(D\) is of size \(1 \times 2\) and \(H\) is of size \(2 \times 2\). Thus, the multiplication \(DH\) is valid because the number of columns in \(D\) (2) matches the number of rows in \(H\) (2).

Step 2: Perform Matrix Multiplication for DH

Since the multiplication is valid, we can compute \(DH\).Let \(D = \begin{bmatrix} 7 & 3 \end{bmatrix}\) and \(H = \begin{bmatrix} 3 & 1 \ 2 & -1 \end{bmatrix}\).The resulting matrix \(DH\) will have dimensions \(1 \times 2\).Compute each element:1. First column: \(7 \times 3 + 3 \times 2 = 21 + 6 = 27\).2. Second column: \(7 \times 1 + 3 \times (-1) = 7 - 3 = 4\).Thus, \(DH = \begin{bmatrix} 27 & 4 \end{bmatrix}\).

Step 3: Check Matrix Multiplication Validity for HD

Now, consider the multiplication \(HD\).Matrix \(H\) is of size \(2 \times 2\) and \(D\) is of size \(1 \times 2\). The multiplication \(HD\) is not valid because the number of columns in \(H\) (2) does not match the number of rows in \(D\) (1).

Step 4: Conclude the Results

Based on the previous steps:(a) The operation \( DH \) is defined and results in \( \begin{bmatrix} 27 & 4 \end{bmatrix}\).(b) The operation \( HD \) is undefined and cannot be performed due to a mismatch in matrix dimensions.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is key to mastering matrix multiplication. Every matrix is defined by its size or dimensions, which is written as \(x \times y\), where \(x\) is the number of rows and \(y\) is the number of columns.

Here's how you describe the dimensions:

• Matrix \(A\) with 2 rows and 2 columns is a \(2 \times 2\) matrix.
• Matrix \(B\) has 2 rows and 3 columns, making it a \(2 \times 3\) matrix.
• Notice that matrices can have different shapes like the \(1 \times 2\) of \(D\) and the \(2 \times 2\) of \(H\).

For matrix multiplication, you can only multiply two matrices if the number of columns in the first matrix matches the number of rows in the second matrix. This is called compatibility. For example, for multiplying \(D \, H\), the first matrix \(D\) has 2 columns, and \(H\) has 2 rows, making it possible.

However, for \(H \, D\), the dimensions aren't compatible, so the operation cannot be performed. Understanding dimensions is the first step in determining the possibility of matrix operations.

Matrix Algebra

Matrix algebra includes operations such as addition, subtraction, and multiplication. Multiplication is particularly interesting as it follows specific rules compared to regular arithmetic.

In matrix multiplication, the product of two matrices results in a new matrix. This isn't straightforward addition or multiplication of individual terms but involves computing each entry by performing a series of multiplications and summations.

Take the multiplication of matrices \(D\) and \(H\) as an example:

• Compute the product by multiplying each element of the row of \(D\) with each element of the columns of \(H\), then summing the resulting products for each position in the resulting matrix.
• For matrix \(D \, H\):
  - First element is calculated as \(7 \times 3 + 3 \times 2 = 21 + 6 = 27\)
  - Second element is \(7 \times 1 + 3 \times (-1) = 7 - 3 = 4\)

The resulting matrix is \( \begin{bmatrix} 27 & 4 \end{bmatrix} \) which is a \(1 \times 2\) matrix. Remember, the order of matrices is essential: \(D \, H\) yields a defined result, but reversing them in \(H \, D\) does not.

Matrix Validity

Matrix validity in multiplication revolves around checking the dimensions. Validity ensures that the operations you're attempting are mathematically possible.

The general rule for matrix multiplication is:

• If matrix \(X\) is \(m \times n\) and matrix \(Y\) is \(n \times p\), multiplication \(X \, Y\) is only valid if the number of columns in \(X\) equals the number of rows in \(Y\).

In our example, \(D = \begin{bmatrix} 7 & 3 \end{bmatrix}\) and \(H = \begin{bmatrix} 3 & 1 \ 2 & -1 \end{bmatrix} \), \(D \, H\) is valid because \(D\) has 2 columns and \(H\) has 2 rows.

On the contrary, \(H \, D\) isn't valid. \(H\) has 2 columns, but \(D\) only has 1 row, leading to a compatibility mismatch. Each matrix operation relies on these checks for correct execution. Always ensure matrix suitability to avoid invalid operations.