Problem 45 - Algebra 2

Question

Find each product, if possible. \(\left[\begin{array}{ll}{2} & {5}\end{array}\right] \cdot\left[\begin{array}{rr}{3} & {1} \\ {-2} & {6}\end{array}\right]\)

Step-by-Step Solution

Answer

The resulting product is \( \left[ \begin{array}{cc} -4 & 32 \end{array} \right] \)."

Step 1: Determine Matrix Dimensions

The first matrix is a 1x2 matrix and the second is a 2x2 matrix. We need to check if the multiplication is possible by confirming the inner dimensions are the same. For a matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, both these value are 2, so multiplication is possible.

Step 2: Perform Matrix Multiplication

To multiply the matrices, we use the row of the first matrix and the columns of the second matrix. Calculate the product as follows: \ Multiply the first row of the first matrix by the first column of the second matrix: \(2 \cdot 3 + 5 \cdot (-2) = 6 - 10 = -4\)\ Multiply the first row of the first matrix by the second column of the second matrix: \(2 \cdot 1 + 5 \cdot 6 = 2 + 30 = 32\)

Step 3: Form the Resulting Matrix

Now, arrange these calculated values into a single 1x2 matrix, because we multiplied a 1x2 matrix by a 2x2 matrix. The resulting matrix is \( \left[ \begin{array}{cc} -4 & 32 \end{array} \right] \).

Key Concepts

Matrix Dimensions

When discussing matrices, understanding matrix dimensions is crucial for operations like multiplication. A matrix's dimensions are described by the number of its rows and columns, denoted as "rows x columns." For example, a matrix with 1 row and 2 columns is a "1x2" matrix.

Matrix multiplication requires matching the inner dimensions of two matrices. Specifically, the number of columns in the first matrix must match the number of rows in the second matrix. If you have a 1x2 matrix and a 2x2 matrix, as in our exercise, multiplication is feasible because the "2" in 1x2 matches the "2" in 2x2. This ensures that every element of the row in the first matrix interacts with every element of the column in the second.

• First Matrix: 1x2 (1 row, 2 columns).
• Second Matrix: 2x2 (2 rows, 2 columns).

If these conditions are met, you can proceed to multiply the matrices confidently.

Resulting Matrix

Once you agree the inner dimensions of the matrices allow for multiplication, it's time to create the resulting matrix. The dimensions of the resulting matrix are determined by the number of rows in the first matrix and the number of columns in the second matrix. Therefore, a 1x2 matrix multiplied by a 2x2 matrix will yield a 1x2 matrix.

Here's how you can find it:

• Resulting matrix dimensions: The resulting matrix from this operation will have 1 row and 2 columns because the first matrix has 1 row, and the second matrix has 2 columns.

We used the row from the first matrix to interact with each column in the second matrix, thus producing a single row in the resulting matrix, signifying the result of our calculations.

After calculating, the resulting matrix we formed is:\[\begin{bmatrix} -4 & 32 \end{bmatrix}\]

Matrix Product

The matrix product is the computed result when two matrices are multiplied according to matrix multiplication rules. To find the product, take a row from the first matrix and multiply it across each column in the second matrix. Sum these products to form elements of the resulting matrix. In matrix multiplication, each element of the resulting matrix is a sum of products. Here's how you can perform it in our case:

• First element (-4): Multiply the first element of the row with the first element of the column, and do the same with the second elements, then add: \(2 \cdot 3 + 5 \cdot (-2) = 6 - 10 = -4\)
• Second element (32): Repeat the process for the second column: \(2 \cdot 1 + 5 \cdot 6 = 2 + 30 = 32\)

The final matrix product consolidates all these calculations into a neat format, directly reflecting the interaction between the rows and columns of the original matrices. Understanding this process brings clarity to the concept of matrix multiplication and how the resulting values are not just shuffled around numbers; they represent precise calculated interactions based on the rules of linear algebra.