Problem 10 - Precalculus: Mathematical for Calculus

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: Determine Dimensions

The first matrix is a 3x2 matrix, and the second matrix is a 2x1 matrix. The rule for matrix multiplication is that the number of columns in the first matrix must equal the number of rows in the second matrix.

Step 2: Check Multiplicability

Since the number of columns in the first matrix (2) matches the number of rows in the second matrix (2), these two matrices can be multiplied.

Step 3: Perform Matrix Multiplication

Multiply the matrices by taking the dot product of the rows of the first matrix with the column of the second matrix.1. First row: \(2*5 + (-3)*1 = 10 - 3 = 7\)2. Second row: \(0*5 + 1*1 = 0 + 1 = 1\)3. Third row: \(1*5 + 2*1 = 5 + 2 = 7\)

Step 4: Write Resultant Matrix

The resulting matrix from the multiplication is a 3x1 matrix:\[\begin{bmatrix} 7 \ 1 \ 7 \end{bmatrix}\]

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial when dealing with matrix multiplication. A matrix is essentially an array of numbers arranged in rows and columns. The dimensions of a matrix are described by the number of rows and columns it has.
For example, if a matrix has 3 rows and 2 columns, like this example:

• The matrix would be called a 3x2 matrix (pronounced as "three by two").
• The first number (3) refers to the number of rows.
• The second number (2) refers to the number of columns.

In contrast, a second matrix with 2 rows and 1 column would be labeled as a 2x1 matrix. To multiply two matrices, an important condition must be met. The number of columns in the first matrix must match the number of rows in the second matrix. In our example, since the first matrix is 3x2 and the second is 2x1, multiplication is possible. This fundamental rule ensures the dot product can be calculated, leading to successful matrix multiplication.

Dot Product

The dot product is a key concept in matrix operations, especially in multiplication. To compute the dot product, you must multiply corresponding elements of two vectors and then sum these products. When we multiply matrices, we use the dot product to combine rows from the first matrix and columns from the second.
In our exercise, each row from the 3x2 matrix is combined with the single column from the 2x1 matrix:

• First row calculation: Multiply the elements of the first row by corresponding elements of the column: \( 2 \times 5 + (-3) \times 1 = 10 - 3 = 7 \).
• Second row calculation: Here, the dot product is \( 0 \times 5 + 1 \times 1 = 0 + 1 = 1 \).
• Third row calculation: Finally, calculate as \( 1 \times 5 + 2 \times 1 = 5 + 2 = 7 \).

This results in each value of the resulting matrix. By understanding the dot product, the process of matrix multiplication becomes clearer and more intuitive.

Matrix Operation

Matrix operations encompass various methods for manipulating matrices, with multiplication being a commonly used operation. In matrix multiplication, you're essentially transforming two matrices into a third matrix by applying the dot product across different combinations of rows and columns.
To guide this process, always remember:

• Matching Dimensions: As mentioned earlier, for matrix multiplication, columns of the first must equal rows of the second matrix.
• Result Dimension: The resulting matrix takes the number of rows from the first matrix and the number of columns from the second matrix. In this case, the result is a 3x1 matrix.
• Sequential Multiplication: The operation must follow systematically, processing each row and column pair individually via the dot product.

Matrix multiplication is not commutative, meaning the order of matrices is vital. This operation is foundational in numerous fields like computer graphics, physics simulations, and more, providing tools for transforming and interpreting complex data sets efficiently.