Problem 73 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The product \( AB \) is not defined due to incompatible dimensions.

Step 1: Check dimensions

\( A \) is \( 2 \times 2 \), \( B \) is \( 3 \times 2 \).

Step 2: Conclusion

For \( AB \), need columns of \( A \) (2) = rows of \( B \) (3). Since \( 2 \neq 3 \), \( AB \) is not defined.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial when it comes to operations involving matrices, like matrix multiplication. A matrix is essentially an array of numbers arranged in rows and columns. The dimensions of a matrix are given in the format of "rows by columns."
For example, if you have a matrix with 2 rows and 3 columns, its dimensions are written as \(2 \times 3\). Knowing the dimensions helps us determine if two matrices can be multiplied together.

• If a matrix has dimensions \(m \times n\), it means it has \(m\) rows and \(n\) columns.
• To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix.

This rule ensures that each element of the first matrix has a corresponding element in the second matrix with which to multiply. If this condition is not met, the multiplication cannot proceed.

Matrix Product

The matrix product, or matrix multiplication, involves a systematic way of multiplying two matrices. Not every pair of matrices can be multiplied, and that's largely because of their dimensions (as discussed earlier).

Here's how the matrix multiplication works:

• Given matrices \(A\) and \(B\), where the number of columns in \(A\) matches the number of rows in \(B\).
• The element in the resulting matrix, located at position \((i, j)\), is obtained by multiplying the elements of the \(i\)-th row of \(A\) and the \(j\)-th column of \(B\), and then summing the results.

This process combines information from both matrices to produce a new matrix. Think of it as combining parts to form a new whole that reflects features of each original matrix.

Undefined Matrix Operation

An undefined matrix operation occurs when conditions necessary for a certain matrix operation are not satisfied. In the case of matrix multiplication, the operation is undefined when the number of columns in the first matrix does not match the number of rows in the second matrix.

For example, consider matrices \(A\) and \(B\):

• Matrix \(A\) has dimensions \(2 \times 2\).
• Matrix \(B\) has dimensions \(3 \times 2\).

To multiply these matrices, the number of columns in \(A\), which is 2, should match the number of rows in \(B\), which is 3. Since they do not match, the operation \(AB\) is undefined.
This concept reminds us of the importance of always checking matrix dimensions before attempting operations, ensuring that our mathematical endeavours are valid and feasible.