Problem 74 - A Graphical Approach to College Algebra

Question

Given \(A=\left[\begin{array}{cc}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{cc}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{ccc}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product if possible. $$C A$$

Step-by-Step Solution

Answer

The product \(CA\) is not possible.

Step 1: Verify Dimensions

To multiply matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix \(C\) is a \(2\times 3\) and matrix \(A\) is a \(2\times 2\). These matrices cannot be multiplied because the number of columns in \(C\) (3) does not match the number of rows in \(A\) (2).

Step 2: Conclusion: Matrix Product not Possible

Since the dimensions do not align for the product \(CA\), the matrix multiplication is not possible.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial in linear algebra, especially when it comes to matrix multiplication. A matrix is defined by its dimensions, which are specified as "number of rows \( \times \) number of columns". For example, a matrix with 2 rows and 3 columns is a \(2 \times 3\) matrix.

When considering the multiplication of two matrices, the dimensions play a key role. A necessary condition for two matrices to be multiplied is that the number of columns in the first matrix should match the number of rows in the second matrix. For example, to multiply a \(2 \times 3\) matrix by a \(3 \times 2\) matrix, the "3" in \(2 \times 3\) (columns) must match the "3" in \(3 \times 2\) (rows).

• If the inner dimensions match, the matrices can be multiplied.
• The resulting matrix will have the "outer dimensions" of the original matrices.

In the provided exercise, the inability to multiply matrix \(C\) (\(2 \times 3\)) and matrix \(A\) (\(2 \times 2\)) arises due to mismatched inner dimensions (3 versus 2). As such, the matrix product \(CA\) is not possible.

Matrix Product

The matrix product is a fundamental operation in linear algebra, reflecting how two matrices interact. When the dimensions align correctly, the matrix product is calculated through a series of dot products.

Here's a simple process to calculate the matrix product:

• Align two matrices such that the number of columns in the first and the number of rows in the second are equal.
• The element in the resulting matrix is found by taking the dot product of a row from the first matrix and a column from the second matrix.
• This means multiplying corresponding elements and summing the products.

This dot product and sum process is repeated for each row-column pair, producing the final matrix.

However, when dimensions don't align, as seen with \(C\) and \(A\), the matrix product cannot be performed. The process won’t begin because there are no pairable rows and columns.

Linear Algebra

Linear algebra is a vast and vital area of mathematics focused on vector spaces and linear mappings between these spaces. Matrices serve as one of the primary tools in representing linear transformations and solving equations.

Matrices are used to organize data or represent systems of linear equations. The operations, including addition, subtraction, and most significantly, multiplication, allow for manipulating and transforming this data.

• Matrix multiplication is particularly useful in finding solutions to systems of equations.
• It can also represent multiple linear transformations compactly.

In practical applications, linear algebra problems often involve verifying compatibility of matrix dimensions, as illustrated in the exercise with matrices \(C\) and \(A\). When the matrix product is not possible due to incompatible dimensions, explorations within linear algebra may include adjusting matrices that can be factored or reordered to permit multiplication in a different context.

Understanding these concepts allows solving complex problems in science, engineering, and computer graphics, showcasing the power of linear algebra beyond its theoretical roots.