Problem 16 - Algebra 2

Question

Determine whether each matrix product is defined. If so, state the dimensions of the product. \(P_{1 \times 3} \cdot Q_{4 \times 1}\)

Step-by-Step Solution

Answer

The matrix product is not defined as the dimensions do not satisfy multiplication conditions.

Step 1: Understand Matrix Dimensions

Matrix dimensions are expressed as rows by columns. The given matrices are \(P_{1 \times 3}\) which means it has 1 row and 3 columns, and \(Q_{4 \times 1}\) which means it has 4 rows and 1 column.

Step 2: Verify Matrix Multiplication Condition

For two matrices to be multiplied, the number of columns in the first matrix (\(P\)) must equal the number of rows in the second matrix (\(Q\)). Here, \(P\) has 3 columns and \(Q\) has 4 rows, so the condition is not satisfied.

Step 3: Conclusion

Since the number of columns in \(P\) does not equal the number of rows in \(Q\), the matrix product \(P \cdot Q\) is not defined.

Key Concepts

Matrix Dimensions

Matrix dimensions are a key component in understanding matrix operations. Dimensions are given in the format of rows by columns. For instance, a matrix labeled as \(P_{1 \times 3}\) denotes a matrix with 1 row and 3 columns. Meanwhile, a matrix labeled as \(Q_{4 \times 1}\) indicates 4 rows and 1 column. Knowing how to read and interpret these labels is essential when considering any matrix operation.
Matrix dimensions dictate not only the size but also the potential operations that can be performed on the matrix. When given two matrices, it's important to first evaluate their dimensions before attempting any operation like addition, subtraction, or multiplication. Only then can you determine how, or if, they can be combined.

Matrix Product Conditions

To multiply two matrices, specific conditions must be met regarding their dimensions. The most important rule is that the number of columns in the first matrix must equal the number of rows in the second matrix. This condition ensures that the multiplication process is possible.
In the exercise, we have matrix \(P_{1 \times 3}\) and matrix \(Q_{4 \times 1}\). Here, the first matrix has 3 columns, and the second matrix has 4 rows. These numbers do not match, which means multiplying these matrices is not possible under standard matrix multiplication rules.
If this condition is met, the resulting matrix will have dimensions defined by the number of rows of the first matrix and the number of columns of the second matrix. If not met, as in \(P \cdot Q\), the matrices cannot be multiplied, and thus the product is undefined.

Matrix Algebra

Matrix algebra involves various operations, primarily addition, subtraction, and multiplication. These operations follow specific rules and conditions based on matrix dimensions.

• Addition and Subtraction: Two matrices can only be added or subtracted if they are of the same dimension. This means the matrices need to have the same number of rows and columns.
• Multiplication: For matrix multiplication, as discussed, the number of columns in the first matrix must match the number of rows in the second. If this condition is satisfied, we can proceed to multiply the matrices.
• Transpose and Inverse: Other operations like transposing (flipping a matrix over its diagonal) and finding an inverse (only possible for square matrices) also exist in matrix algebra.

Understanding these algebraic operations is fundamental in fields such as computer science, physics, and statistics where matrices are widely applied. Mastering the basic rules helps in executing more complex mathematical computations and solving practical problems with matrices.