Problem 10 - Algebra and Trigonometry

Question

Perform the matrix operation, or if it is impossible, explain why. $$ \left[\begin{array}{rr}{2} & {-3} \\ {0} & {1} \\ {1} & {2}\end{array}\right]\left[\begin{array}{l}{5} \\ {1}\end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix is \(\begin{bmatrix} 7 \\ 1 \\ 7 \end{bmatrix}\).

Step 1: Identify Matrix Dimensions

The first matrix is of dimensions 3x2 (3 rows and 2 columns), and the second matrix is of dimensions 2x1 (2 rows and 1 column). Matrix multiplication is possible if the number of columns in the first matrix equals the number of rows in the second matrix. Here, both are equal to 2, so multiplication is possible.

Step 2: Set Up the Multiplication

To perform the multiplication, take each row of the first matrix and multiply it by the column of the second matrix. This involves taking the dot product of each row with the column.

Step 3: Multiply First Row by the Column Matrix

The first row of the first matrix is \([2, -3]\). Multiply this by the column \([5, 1]\) as follows: \(2\times5 + (-3)\times1 = 10 - 3 = 7\).

Step 4: Multiply Second Row by the Column Matrix

The second row of the first matrix is \([0, 1]\). Multiply this by the column \([5, 1]\) as follows: \(0\times5 + 1\times1 = 0 + 1 = 1\).

Step 5: Multiply Third Row by the Column Matrix

The third row of the first matrix is \([1, 2]\). Multiply this by the column \([5, 1]\) as follows: \(1\times5 + 2\times1 = 5 + 2 = 7\).

Step 6: Compile the Resulting Matrix

Combine the results from each row's multiplication to form the resulting matrix: \[\left[\begin{array}{c}{7} \ {1} \ {7}\end{array}\right]\]. This is a 3x1 matrix.

Key Concepts

Matrix Operations

Matrix operations are a fundamental aspect of mathematics, especially in fields like computer science and engineering. They allow us to manipulate, transform, and combine matrices to extract useful information. One crucial matrix operation is multiplication. To multiply matrices, each element in a row of the first matrix is multiplied by the corresponding element in a column of the second matrix. The results are then summed up to produce an element in the resulting matrix. This operation is known as the dot product, which works only under certain conditions that depend on the dimensions of the matrices being multiplied. Understanding these operations is key to working with matrices effectively.

Matrix Dimensions

Matrix dimensions are essential when performing operations like multiplication. A matrix dimension is described in terms of its rows and columns, generally in the format of 'rows x columns'.
For example, a matrix with 3 rows and 2 columns is a 3x2 matrix.

• The number of columns in the first matrix must equal the number of rows in the second matrix to multiply them. This requirement ensures that the dot products can be calculated.
• If these dimensions do not align, matrix multiplication is not possible, and we need to revise the matrices to find a compatible pairing.

Understanding the concept of dimensions can save time and facilitate accurate results when dealing with complex matrix operations.

Dot Product

The dot product is a crucial operation when multiplying matrices. It is the backbone of generating each entry in the resulting matrix during multiplication. For each element of the resulting matrix, you take one row from the first matrix and a column from the second matrix.
Here’s how it works:

• Multiply each corresponding element from the row and column.
• Add all these products together to get the single number, or element, of the resulting matrix.

For example, consider a row \([2, -3]\) from the first matrix and a column \([5, 1]\) from the second matrix. Compute: \(2 \times 5 + (-3) \times 1 = 10 - 3 = 7\).
This calculation results in one element of the final matrix. This method, repeated for each row of the first matrix, allows the complete resulting matrix to be formed.

Resulting Matrix

The resulting matrix is the final product of a matrix multiplication operation. For each row in the first matrix multiplied by the column in the second matrix, an element in the resulting matrix is formed. The size of the resulting matrix depends on the number of rows from the first matrix and columns from the second matrix.
For example, if you multiply a 3x2 matrix with a 2x1 matrix, the resulting matrix is a 3x1 matrix because:

• It inherits the number of rows from the first matrix.
• It inherits the number of columns from the second matrix.

In the exercise given, the calculated elements \([7, 1, 7]\) form the resulting matrix. Skills in determining the dimensions and calculating each element are essential for constructing the correct final matrix after the multiplication process.