Problem 24 - Algebra 2

Question

Find each product, if possible. \(\left[\begin{array}{rrr}{4} & {-1} & {6} \\ {1} & {5} & {-8}\end{array}\right] \cdot\left[\begin{array}{rr}{1} & {3} \\ {9} & {-6}\end{array}\right]\)

Step-by-Step Solution

Answer

Matrix multiplication is not possible due to dimension mismatch.

Step 1: Determine Matrix Compatibility

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. In this case, the first matrix is a 2x3 matrix and the second is a 2x2 matrix. Since 3 (columns of the first) does not equal 2 (rows of the second), matrix multiplication cannot be performed.

Step 2: Conclusion

Since the matrices are not compatible for multiplication, the product cannot be calculated. We cannot proceed with the multiplication due to the mismatch in dimensions.

Key Concepts

Matrix Compatibility

When we talk about matrix compatibility for multiplication, we're referring to a specific condition that must be met in order to multiply two matrices. This condition is based on the number of columns and rows that the matrices have.
To check if two matrices are compatible for multiplication:

• The number of columns in the first matrix must be the same as the number of rows in the second matrix.

If this condition is not met, you cannot multiply the matrices together, and therefore, a product will not exist.
In our original exercise, the first matrix is a 2x3 matrix, which means it has 2 rows and 3 columns.
The second matrix is a 2x2 matrix, meaning it has 2 rows and 2 columns.
We can immediately see that the number of columns in the first matrix (3) does not match the number of rows in the second matrix (2).
Thus, these matrices are not compatible for multiplication.

Matrix Dimensions

Understanding matrix dimensions is crucial for performing operations like multiplication in linear algebra.
Matrix dimensions tell us the size of a matrix, and they are usually expressed in terms of the number of rows by the number of columns:

• A matrix with 2 rows and 3 columns is called a 2x3 matrix.
• A matrix with 3 rows and 4 columns is a 3x4 matrix.

When working with matrices, always check their dimensions first.
In the context of our exercise, recognizing that one matrix is 2x3 and the other is 2x2 helps us identify that they are incompatible for multiplication, as previously explained.
Having a clear understanding of matrix dimensions allows you to quickly assess whether multiplication is possible.
It also aids in planning how the resultant matrix might look if multiplication is possible, since the product will then have dimensions of the outer numbers of the two compatible matrices.

Matrix Multiplication Rules

Matrix multiplication is not like simple multiplication we perform with numbers. It involves a specific set of rules and steps:
Before you can even start the multiplication process, you must ensure that the matrices are compatible for multiplication.
If they are, here’s how to proceed:

• Take each row of the first matrix and multiply it by each column of the second matrix.
• The element in the new matrix, which is the result, is found by taking the sum of these products.

For instance, an element in the resulting matrix is the sum of products of corresponding elements from a row of the first and a column of the second matrix.
This means that each entry in the result is a dot product. If the matrices aren’t compatible, like in the given exercise, you won't be able to compute a result at all because the dot products cannot be calculated.
Understanding and remembering these rules helps you apply matrix multiplication correctly in any scenario.