Problem 73 - 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. $$A B$$

Step-by-Step Solution

Answer

The product \(A \times B\) is not possible.

Step 1: Check dimensions

Matrix \( A \) is \( 2 \times 2 \) and matrix \( B \) is \( 3 \times 2 \).

Step 2: Check compatibility

For the product \( AB \) to exist, the number of columns of \( A \) must equal the number of rows of \( B \). Here, columns of \( A = 2 \) and rows of \( B = 3 \). Since \( 2 \neq 3 \), the product \( AB \) is not defined.

Key Concepts

Matrix Dimensions

When working with matrix multiplication, understanding matrix dimensions is extremely important. A matrix is defined by its dimensions, which refers to the number of rows and columns it contains. For example, a matrix with 2 rows and 3 columns is termed a "2x3 matrix". In general, a matrix of size \(m\times n\) has \(m\) rows and \(n\) columns.

The ability to multiply two matrices depends on these dimensions matching in a specific way. For two matrices, say Matrix A of size \(m\times n\) and Matrix B of size \(p\times q\), multiplication is only possible when the number of columns in Matrix A matches the number of rows in Matrix B (i.e., \(n = p\)).

• Check if the number of columns in the first matrix equals the number of rows in the second matrix.
• If they match, multiplication can proceed. Otherwise, it's undefined.
• The resulting product will have dimensions given by the outer dimensions: \(m\times q\).

Matrix Product

The result of multiplying two matrices is known as a matrix product. It's important to know how this product is computed.

When you multiply matrix \(A\) with dimensions \(m x n\) by matrix \(B\) with dimensions \(n x p\), the resulting matrix product will be of dimensions \(m x p\).

To compute each element in the product, follow these steps:

• Select the row from the first matrix and the column from the second matrix.
• Multiply the elements of the row by the corresponding elements of the column.
• Sum all these products to get the single number that goes into the corresponding position in the resulting matrix.

Understanding this element-wise multiplication process is essential for solving matrix multiplication problems. For example, \(A_{i,j}\) in the resulting product is calculated by summing the products of corresponding elements in the individual row of \(A\) and column of \(B\). However, remember that this only works when the mentioned dimension condition is met.

Matrix Arithmetic

Matrix arithmetic involves operations such as addition, subtraction, and multiplication of matrices. While addition and subtraction can be straightforward, as they only require the matrices to have the exact same dimensions, multiplication is a bit stricter in its rules.

Here's a succinct look at the rules for basic matrix arithmetic:

• Addition/Subtraction: Only possible with matrices of the same size. Add or subtract corresponding entries from both matrices.
• Multiplication: More complex, requires the number of columns in the first matrix to match the number of rows in the second matrix.
• Scalar Multiplication: Multiply every element of the matrix by the scalar (a constant number).

When considering matrix multiplication as part of matrix arithmetic, each element in the resultant matrix comes from summing products of entries. This process showcases the versatility and importance of matrices in solving linear equations and transformations in various fields.