Problem 13 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The matrices cannot be multiplied due to dimension incompatibility.

Step 1: Determine the Dimensions of the Matrices

First, identify the dimensions of the matrices. The first matrix is a 3x2 matrix and the second matrix is a 3x2 matrix. The dimensions are given as m x n, where m is the number of rows and n is the number of columns.

Step 2: Check Compatibility for Multiplication

For two matrices to be multiplied, the number of columns of the first matrix must equal the number of rows of the second matrix. Here, the first matrix has 2 columns and the second matrix has 3 rows. Since 2 ≠ 3, these matrices cannot be multiplied.

Step 3: Conclusion on Matrix Multiplication

Since the number of columns in the first matrix does not match the number of rows in the second, the operation is impossible. Therefore, these matrices cannot be multiplied.

Key Concepts

Matrix Dimensions

Matrix dimensions are the foundational blocks when dealing with any matrix operation. A matrix is a rectangular array of numbers arranged in rows and columns. The size or dimension of a matrix is expressed as 'm by n' (written as \( m \times n \)), where \( m \) is the number of rows and \( n \) is the number of columns. Understanding dimensions is crucial because it dictates many properties of the matrix, including its compatibility for operations like multiplication.
For example, consider a matrix with three rows and two columns:

• The matrix dimensions are \( 3 \times 2 \).
• This means there are 3 individual rows, each containing 2 elements.

Knowing these dimensions is the first step in performing matrix operations, as the dimensions decide how matrices interact with each other.

Matrix Compatibility

Matrix compatibility checks are a key step before commencing any matrix multiplication. 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. This specific requirement is integral because each row of the first matrix will eventually combine with each column of the second matrix to produce the resultant matrix.
To illustrate, given a \( 3 \times 2 \) matrix and attempting to multiply it with another \( 3 \times 2 \) matrix:

• The first matrix has 2 columns.
• The second matrix has 3 rows.

In this case, since 2 (columns of the first matrix) is not equal to 3 (rows of the second matrix), the matrices cannot be multiplied. This compatibility rule helps in understanding why certain operations are feasible and others are not, based purely on matrix dimensions.

Matrix Operations

Matrix operations encompass a variety of tasks such as addition, subtraction, and particularly multiplication. Matrix multiplication is a more complex operation than simple arithmetic with numbers. It involves the pairing of elements across the rows of the first matrix with the columns of the second matrix. A successful multiplication results in a new matrix, often with different dimensions from the original matrices.
When attempting matrix multiplication, after confirming compatibility:

• The number of rows in the result is from the first matrix.
• The number of columns in the result is from the second matrix.

This ensures the operation aligns correctly to form a meaningful product matrix. Due to its complexity, using matrix operations effectively requires careful attention to detail and adherence to existing rules, most importantly the understanding and verification of matrix compatibility first. Matrix operations can unlock a world of possibilities in solving complex problems, provided the fundamental rules of multiplication are followed.