Problem 8 - Algebra and Trigonometry

Question

Perform the matrix operation, or if it is impossible, explain why. $$ \left[\begin{array}{lll}{2} & {1} & {2} \\ {6} & {3} & {4}\end{array}\right]\left[\begin{array}{rr}{1} & {-2} \\ {3} & {6} \\ {-2} & {0}\end{array}\right] $$

Step-by-Step Solution

Answer

The product of the matrices is \( \left[\begin{array}{cc} 1 & 2 \\ 7 & 6 \end{array}\right] \).

Step 1: Identify the Matrices and Their Dimensions

First, let's identify the given matrices and determine their dimensions. The first matrix \( A \) is \( 2 \times 3 \), as it has 2 rows and 3 columns: \[A = \left[\begin{array}{ccc} 2 & 1 & 2 \ 6 & 3 & 4 \end{array}\right] \]The second matrix \( B \) is \( 3 \times 2 \), as it has 3 rows and 2 columns: \[B = \left[\begin{array}{rr} 1 & -2 \ 3 & 6 \ -2 & 0 \end{array}\right] \]

Step 2: Determine the Possibility of Multiplication

Matrix multiplication is possible when the number of columns in the first matrix is equal to the number of rows in the second matrix. For matrix \( A \), there are 3 columns, and for matrix \( B \), there are 3 rows. Thus, it is possible to multiply these matrices as the inner dimensions are equal (3).

Step 3: Compute the Resultant Matrix

The result of multiplying a \( 2 \times 3 \) matrix by a \( 3 \times 2 \) matrix is a \( 2 \times 2 \) matrix. Let's compute this:- **First row, first column:** \((2 \times 1) + (1 \times 3) + (2 \times -2) = 2 + 3 - 4 = 1 \)- **First row, second column:** \((2 \times -2) + (1 \times 6) + (2 \times 0) = -4 + 6 + 0 = 2 \)- **Second row, first column:** \((6 \times 1) + (3 \times 3) + (4 \times -2) = 6 + 9 - 8 = 7 \)- **Second row, second column:** \((6 \times -2) + (3 \times 6) + (4 \times 0) = -12 + 18 + 0 = 6 \)The resultant matrix is:\[C = \left[\begin{array}{cc} 1 & 2 \ 7 & 6 \end{array}\right]\]

Step 4: Verify the Solution

Double-check the matrix operations to ensure the calculations were done correctly for each entry in matrix \( C \). Recalculate if errors are found. The computed entries should match: First entry (1), second entry (2), third entry (7), and fourth entry (6). The computation has been verified.

Key Concepts

Matrix Dimensions

When dealing with matrices, understanding their dimensions is crucial. The dimension of a matrix is defined by the number of rows and columns it contains. In this case, the matrix is often described by 'rows by columns.' For example, if we have a matrix with 2 rows and 3 columns, we say it is a \(2 \times 3\) matrix.

Knowing the dimensions helps us understand what operations we can perform with the matrices. In our exercise, we had two matrices. The first matrix was a \(2 \times 3\) matrix, meaning it had 2 rows and 3 columns:

• Matrix A: \[ \left[\begin{array}{ccc} 2 & 1 & 2 \ 6 & 3 & 4 \end{array}\right] \]

For the second matrix, which is a \(3 \times 2\) matrix, it had 3 rows and 2 columns:

• Matrix B: \[ \left[\begin{array}{rr} 1 & -2 \ 3 & 6 \ -2 & 0 \end{array}\right] \]

Understanding these dimensions is the first step to knowing if we can multiply two matrices.

Matrix Operations

Matrix operations, such as addition, subtraction, and multiplication, require careful attention to the dimensions of the matrices involved. For addition and subtraction, the matrices must have the same dimensions.

Multiplication is more complex. Matrix multiplication can be performed if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. This requirement is crucial for the operation to be defined. In this scenario, matrix \( A \) is \( 2 \times 3 \) and matrix \( B \) is \( 3 \times 2 \). The inner dimensions (3 for both matrices) are equal, allowing multiplication to proceed.

The multiplication process involves taking each row from the first matrix and combining it with each column from the second matrix, which can be visualized as a dot product of the vectors. This operation outputs a new matrix, which brings us to the resultant matrix.

Resultant Matrix

The resultant matrix is the outcome of multiplying two matrices together. Once multiplication is determined to be possible, we proceed to calculate the new matrix's entries. The new matrix's dimensions are determined by the outer dimensions of the original matrices. In this exercise, the product of a \( 2 \times 3 \) matrix and a \( 3 \times 2 \) matrix results in a \( 2 \times 2 \) matrix.

To populate the resultant matrix, we calculate each entry individually by taking the sum of the products of corresponding elements from the rows of the first matrix and the columns of the second matrix. For instance:

• First row, first column: \((2 \times 1) + (1 \times 3) + (2 \times -2) = 1\)
• First row, second column: \((2 \times -2) + (1 \times 6) + (2 \times 0) = 2\)
• Second row, first column: \((6 \times 1) + (3 \times 3) + (4 \times -2) = 7\)
• Second row, second column: \((6 \times -2) + (3 \times 6) + (4 \times 0) = 6\)

Thus, the resulting matrix is:
\[C = \left[\begin{array}{cc} 1 & 2 \ 7 & 6 \end{array}\right]\]