Problem 9 - Algebra and Trigonometry

Question

Perform the matrix operation, or if it is impossible, explain why. $$ \left[\begin{array}{rr}{1} & {2} \\ {-1} & {4}\end{array}\right]\left[\begin{array}{rrr}{1} & {-2} & {3} \\ {2} & {2} & {-1}\end{array}\right] $$

Step-by-Step Solution

Answer

The product is a 2x3 matrix: \(\begin{bmatrix} 5 & 2 & 1 \\ 7 & 10 & -7 \end{bmatrix}\).

Step 1: Identify the Dimensions

First, check the dimensions of the two matrices involved in the operation. The first matrix is a 2x2 matrix (2 rows and 2 columns), while the second matrix is a 2x3 matrix (2 rows and 3 columns).

Step 2: Confirm Compatibility

For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Since both conditions hold true (2 columns in the first and 2 rows in the second), the matrices can be multiplied.

Step 3: Set Up the Multiplication

To multiply these matrices, take each row of the first matrix and multiply it against each column of the second matrix. This will result in a matrix with dimensions copying the rows of the first matrix and columns of the second matrix (2x3).

Step 4: Perform Matrix Multiplication

Calculate each entry of the resulting matrix:- First row, first column: \((1 \cdot 1) + (2 \cdot 2) = 1 + 4 = 5\)- First row, second column: \((1 \cdot (-2)) + (2 \cdot 2) = -2 + 4 = 2\)- First row, third column: \((1 \cdot 3) + (2 \cdot (-1)) = 3 - 2 = 1\)- Second row, first column: \((-1 \cdot 1) + (4 \cdot 2) = -1 + 8 = 7\)- Second row, second column: \((-1 \cdot (-2)) + (4 \cdot 2) = 2 + 8 = 10\)- Second row, third column: \((-1 \cdot 3) + (4 \cdot (-1)) = -3 - 4 = -7\).

Step 5: Write the Resulting Matrix

The resulting matrix after completing the multiplication is:\[\begin{bmatrix} 5 & 2 & 1 \7 & 10 & -7\end{bmatrix} \]

Key Concepts

Matrix Dimensions

Understanding the dimensions of matrices is crucial when dealing with matrix multiplication. Each matrix is defined by two numbers that indicate its structure: the number of rows and the number of columns. These numbers are often denoted in an "(m x n)" format, where "m" represents the number of rows and "n" represents the number of columns.

For example, if you look at the original exercise:

• The first matrix is a 2x2 matrix, meaning it has 2 rows and 2 columns.
• The second matrix is a 2x3 matrix, which means it has 2 rows and 3 columns.

These dimensions help us identify how matrices can interact with each other.

Whenever you encounter a matrix multiplication problem, the first step should always be to write down or note the dimensions, as they are integral to determining the next steps.

Matrix Compatibility

Matrix compatibility is a term that describes whether two matrices can be multiplied together. This is determined by matching the appropriate dimensions of the matrices. Specifically, for two matrices to be compatible for multiplication, the number of columns in the first matrix must match the number of rows in the second matrix.

Going back to our example:

• The first matrix has 2 columns, and the second matrix has 2 rows. This condition is satisfied.

Since the condition of compatibility holds true, matrix multiplication can proceed. Understanding matrix compatibility is pivotal because if the condition does not hold, multiplication is impossible. It’s similar to fitting a key into a lock; if the shapes do not match, the door won’t open. In this case, if the dimensions do not match, no multiplication can occur.

Thus, checking compatibility is a vital step before applying matrix multiplication techniques.

Resulting Matrix

Once you've determined the matrix dimensions and checked for compatibility, the next focus is on the resulting matrix after matrix multiplication. The resulting matrix's size is defined by the number of rows from the first matrix and the number of columns from the second matrix.

Considering the example given, where you have a 2x2 matrix and a 2x3 matrix, the resulting matrix will be a 2x3 matrix:

• The resulting matrix inherits the 2 rows from the first matrix and the 3 columns from the second matrix.

This resulting matrix compilation can initially seem confusing, but visualizing or sketching out the grids can help.

Performing the actual multiplication involves taking rows from the first matrix and columns from the second matrix, multiplying their corresponding elements, and summing the results. This process is done for each cell in the resulting matrix.

The matrix we obtained from the multiplication in the original problem is:\[\begin{bmatrix} 5 & 2 & 1 \ 7 & 10 & -7 \end{bmatrix} \]Understanding how the resulting matrix is formed will help you grasp future matrix operations with ease.