Problem 27 - 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. $$ \left(\begin{array}{llll}{\text { a) }} & {A H} & {} & {\text { (b) }} & {H A}\end{array}\right. $$

Step-by-Step Solution

Answer

Both multiplications are possible. \(AH = \begin{pmatrix} -4 & 7 \\ 14 & -7 \end{pmatrix}\), \(HA = \begin{pmatrix} 6 & -8 \\ 4 & -17 \end{pmatrix}\).

Step 1: Determine Matrix Compatibility for AH

Matrix multiplication is possible if the number of columns in the first matrix equals the number of rows in the second matrix. Matrix \(A\) is a \(2 \times 2\) matrix and Matrix \(H\) is also a \(2 \times 2\) matrix. Since the number of columns in \(A\) (2 columns) equals the number of rows in \(H\) (2 rows), the multiplication \(AH\) can be performed.

Step 2: Perform Matrix Multiplication AH

To multiply matrices \(A\) and \(H\), calculate each element of the resulting matrix by taking the dot product of the corresponding rows and columns. The resulting matrix will be a \(2 \times 2\) matrix.\[ AH = \begin{pmatrix} 2 & -5 \ 0 & 7 \end{pmatrix} \begin{pmatrix} 3 & 1 \ 2 & -1 \end{pmatrix} = \begin{pmatrix} (2 \cdot 3 + (-5) \cdot 2) & (2 \cdot 1 + (-5) \cdot (-1)) \ (0 \cdot 3 + 7 \cdot 2) & (0 \cdot 1 + 7 \cdot (-1)) \end{pmatrix} \]After calculation:\[ AH = \begin{pmatrix} 6 - 10 & 2 + 5 \ 0 + 14 & 0 - 7 \end{pmatrix} = \begin{pmatrix} -4 & 7 \ 14 & -7 \end{pmatrix} \]

Step 3: Determine Matrix Compatibility for HA

Consider matrix \(H\) of size \(2 \times 2\) and matrix \(A\) of size \(2 \times 2\). The multiplication \(HA\) is possible because the number of columns in \(H\) (2 columns) equals the number of rows in \(A\) (2 rows).

Step 4: Perform Matrix Multiplication HA

To multiply matrices \(H\) and \(A\), calculate each element of the resulting matrix by taking the dot product of the corresponding rows and columns. The resulting matrix will be a \(2 \times 2\) matrix.\[ HA = \begin{pmatrix} 3 & 1 \ 2 & -1 \end{pmatrix} \begin{pmatrix} 2 & -5 \ 0 & 7 \end{pmatrix} = \begin{pmatrix} (3 \cdot 2 + 1 \cdot 0) & (3 \cdot (-5) + 1 \cdot 7) \ (2 \cdot 2 + (-1) \cdot 0) & (2 \cdot (-5) + (-1) \cdot 7) \end{pmatrix} \]After calculation:\[ HA = \begin{pmatrix} 6 + 0 & -15 + 7 \ 4 + 0 & -10 + (-7) \end{pmatrix} = \begin{pmatrix} 6 & -8 \ 4 & -17 \end{pmatrix} \]

Key Concepts

Dot Product

The dot product is a fundamental operation in linear algebra that is used to calculate the multiplication of matrices. It involves multiplying the corresponding entries of two sequences of numbers and then summing those products. In the context of matrix multiplication, the dot product is used to generate each element of the resulting matrix.
For example, if we are multiplying a row from matrix \( A \) by a column from matrix \( H \), each element of the dot product is calculated as follows:

• Choose corresponding elements from the row of matrix \( A \) and the column of matrix \( H \).
• Multiply these chosen elements together.
• Sum all the resulting products to form a single number: the element of the new matrix position.

For instance, using matrices \( A \) and \( H \) from the exercise, one element of the matrix product \( AH \) could be calculated by taking the dot product of the first row of \( A \) \([2, -5]\) with the first column of \( H \) \([3, 2]\): \[ (2 \cdot 3) + (-5 \cdot 2) = 6 - 10 = -4 \] This completed calculation provides the top-left element of the resulting matrix \( AH \).
The dot product is essential for understanding how multiplication in matrices works, particularly for transforming and projecting data in geometric and sophisticated data spaces.

Matrix Compatibility

Matrix compatibility is an important concept that dictates whether two matrices can be multiplied together. For two matrices to be compatible for multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. When this condition holds, multiplication can proceed, and a resultant matrix is produced.
For instance, in the exercise above with matrices \( A \) and \( H \) both being \( 2 \times 2 \) matrices, they are compatible for multiplication since matrix \( A \) has 2 columns and matrix \( H \) has 2 rows. Therefore, operations like \( AH \) and \( HA \) can be carried out.
When analyzing matrix compatibility, be sure to:

• Identify the dimensions of both matrices involved in the operation.
• Check that the number of columns in the first matrix equals the number of rows in the second matrix.
• Understand that incompatible matrices, those where this core condition does not hold, cannot be multiplied.

Recognizing matrix compatibility ensures correct calculation procedures and avoids unnecessary errors in linear algebra computations.

2x2 Matrix

A \( 2 \times 2 \) matrix is one of the simplest forms of matrices, consisting of 2 rows and 2 columns. These matrices are often used in basic geometric transformations, like rotation or scaling, and are an ideal starting point for students new to matrix multiplication.
For a general \( 2 \times 2 \) matrix \( M \), it can be represented as:\[ M = \begin{pmatrix} a & b \ c & d \end{pmatrix} \]Here, \( a, b, c, \) and \( d \) are the elements of the matrix. In arithmetic operations like addition and multiplication, the elements are used to systematically compute new matrices.
In the realm of matrix multiplication involving \( 2 \times 2 \) matrices, such as \( AH \) or \( HA \) from the exercise:

• Compute four separate dot products to produce the elements of the resulting \( 2 \times 2 \) matrix.
• Each dot product requires precise attention to detail to ensure accuracy.
• The size of the resultant matrix will remain \( 2 \times 2 \) after multiplication since they started as two \( 2 \times 2 \) matrices.

A strong understanding of \( 2 \times 2 \) matrices and their operations lays a solid foundation for tackling more complex matrices and algebraic equations that appear in advanced math and applications.