Problem 15 - Precalculus: Mathematics for Calculus

Question

Perform the matrix operation, or if it is impossible, explain why. $$\left[\begin{array}{rr} 2 & -3 \\ 0 & 1 \\ 1 & 2 \end{array}\right]\left[\begin{array}{l} 5 \\ 1 \end{array}\right]$$

Step-by-Step Solution

Answer

The product is a 3x1 matrix: \( \begin{bmatrix} 7 \\ 1 \\ 7 \end{bmatrix} \).

Step 1: Identify the Dimensions of the Matrices

Begin by noting the dimensions of the matrices involved. The first matrix \( A \) is a \( 3 \times 2 \) matrix:\[A = \begin{bmatrix} 2 & -3 \ 0 & 1 \ 1 & 2 \end{bmatrix}\]The second matrix \( B \) is a \( 2 \times 1 \) matrix:\[B = \begin{bmatrix} 5 \ 1 \end{bmatrix}\]For matrix multiplication to be possible, the number of columns in the first matrix must match the number of rows in the second matrix. Here, both conditions hold as the first matrix has 2 columns and the second matrix has 2 rows.

Step 2: Multiply the Matrices

To multiply the matrices, each element in the resulting matrix is obtained by taking the dot product of the corresponding row of the first matrix and the column of the second matrix.- The first element is obtained by multiplying the first row of \( A \) with \( B \): \[ (2)(5) + (-3)(1) = 10 - 3 = 7 \]- The second element is obtained by multiplying the second row of \( A \) with \( B \): \[ (0)(5) + (1)(1) = 0 + 1 = 1 \]- The third element is obtained by multiplying the third row of \( A \) with \( B \): \[ (1)(5) + (2)(1) = 5 + 2 = 7 \]

Step 3: Write the Resulting Matrix

The resulting matrix from the multiplication is a \( 3 \times 1 \) matrix formed by the results of the dot products:\[\begin{bmatrix} 7 \ 1 \ 7 \end{bmatrix}\]This matrix represents the product of the given matrices.

Key Concepts

Understanding Matrix Dimensions

Matrix dimensions are crucial when dealing with matrix multiplication. Each matrix is defined by its number of rows and columns, expressed in the format \( m \times n \). Here, \( m \) signifies the row count, while \( n \) reflects the column count. This structure determines the matrix's usability in multiplication operations.
In the exercise, we have two matrices: Matrix \( A \) with dimensions \( 3 \times 2 \), and Matrix \( B \) which is \( 2 \times 1 \).

• Matrix \( A \): 3 rows and 2 columns
• Matrix \( B \): 2 rows and 1 column

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. This is why, in this exercise, matrix multiplication is possible, as the dimensions align correctly: the 2 columns in Matrix \( A \) match the 2 rows in Matrix \( B \).
This understanding of dimensions ensures that each element in the resultant matrix is computed accurately by aligning corresponding row and column elements.

Calculating the Dot Product

The dot product is a fundamental operation in matrix multiplication. It is the process of multiplying corresponding elements of a row from the first matrix with the elements of a column from the second matrix, then summing the results.
In our given problem, each element of the resulting matrix comes from the dot product of the rows of Matrix \( A \) with the column of Matrix \( B \).
Here's how each element was calculated:

• First element: Row 1 of \( A \) and column of \( B \): \( 2 \times 5 + (-3) \times 1 = 7 \)
• Second element: Row 2 of \( A \) and column of \( B \): \( 0 \times 5 + 1 \times 1 = 1 \)
• Third element: Row 3 of \( A \) and column of \( B \): \( 1 \times 5 + 2 \times 1 = 7 \)

Understanding the dot product helps in visualizing how two matrices interact during multiplication. It highlights the transformation from the initial matrix pair to a single, comprehensive result.

Interpreting the Matrix Product Result

After performing the dot products for each corresponding row and column pairs, we derive a new matrix, often referred to as the matrix product result.
In this exercise, multiplying Matrix \( A \) by Matrix \( B \), using the dimensions and dot product principles, we achieve a \( 3 \times 1 \) result matrix:
\[\begin{bmatrix} 7 \ 1 \ 7 \end{bmatrix}\]
This new matrix represents the combined effect of the linear transformations represented by the original matrices. Each element shows the net effect on the output by sequentially applying the operations encoded in the original matrices.
Understanding the final result matrix is crucial for interpreting the outcome of matrix multiplications in practical applications, like scientific computations, computer graphics transformations, and in data processing involving transformations.