Problem 52 - Algebra 2

Question

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

Step-by-Step Solution

Answer

Matrix multiplication is not possible due to incompatible dimensions.

Step 1: Check dimensions

The first matrix is \( 2 \times 3 \) and the second is \( 2 \times 2 \).

Step 2: Conclusion

For multiplication, columns of the first (3) must equal rows of the second (2). Since \( 3 \neq 2 \), the product is not possible.

Key Concepts

Matrix Dimensions

Matrix dimensions play a crucial role in determining whether two matrices can be multiplied. A matrix's dimensions are given in terms of rows and columns. For example, a matrix with 2 rows and 3 columns is referred to as a 2x3 matrix. It's important to always note the order, as mistaking rows for columns can lead to errors.

Before multiplying matrices, it's essential to check their dimensions. For successful multiplication, the number of columns in the first matrix must match the number of rows in the second matrix. This alignment is critical to ensure that every element has a matching counterpart for multiplication and addition. In our specific example, the first matrix is a 2x3 matrix, and the second is a 2x2 matrix. Clearly, the column count of the first matrix doesn't match the row count of the second, which sets the stage for issues in matrix multiplication.

Matrix Multiplication Not Possible

Sometimes, matrix multiplication is simply not possible due to mismatched dimensions. As earlier discussed, the alignment of matrix dimensions is key. Whenever the number of columns in the first matrix doesn't equal the number of rows in the second matrix, you cannot perform matrix multiplication.

In our exercise, the first matrix has 3 columns, while the second matrix only has 2 rows. This mismatch in dimensions makes the multiplication operation impossible. Therefore, always carefully examine the dimensions before proceeding with multiplication. Remember that a mismatch not only prevents calculations but also indicates a fundamental misalignment in the intended operation.

Row and Column Consistency

The concept of row and column consistency highlights the importance of aligning matrix dimensions for multiplication. Think of each row in a matrix as a vector that needs a corresponding column vector from another matrix to perform the multiplication.

For the multiplication to proceed, each element in a matrix's row needs a matching element in the corresponding matrix's column. Imagine trying to multiply vectors of different lengths — it simply wouldn't work. This analogy extends to matrices as well.

In the exercise given, the lack of consistency in rows and columns between the matrix dimensions caused the multiplication to be unfeasible. This underlines why checking dimensions and ensuring row-column alignment is always the first step in matrix multiplication. A consistent pairing of rows and columns allows for the seamless execution of matrix arithmetic.