Problem 24 - Precalculus

Question

perform the indicated operations for each expression, if possible. $$A=\left[\begin{array}{rrr}-1 & 3 & 0 \\\2 & 4 & 1\end{array}\right] \quad B=\left[\begin{array}{rrr}0 & 2 & 1 \\\3 & -2 & 4\end{array}\right] \quad C=\left[\begin{array}{rr}0 & 1 \\\2 & -1 \\\3 & 1\end{array}\right] \quad D=\left[\begin{array}{rr}2 & -3 \\\0 & 1 \\\4 & -2\end{array}\right]$$ $$2B-3A$$

Step-by-Step Solution

Answer

The resulting matrix is \(\left[\begin{array}{ccc}3 & -5 & 2 \\ 0 & -16 & 5 \end{array}\right]\).

Step 1: Define the Operation

We need to perform the operation \(2B - 3A\), which involves scalar multiplication of matrices and matrix subtraction. First, understand that matrix \(B\) and \(A\) are both 2x3 matrices, allowing these operations to be valid.

Step 2: Scalar Multiplication of Matrices

Multiply each element in matrix \(B\) by 2. \[ 2B = \left[ \begin{array}{ccc}2 \times 0 & 2 \times 2 & 2 \times 1 \2 \times 3 & 2 \times (-2) & 2 \times 4 \\end{array} \right] = \left[ \begin{array}{ccc}0 & 4 & 2 \6 & -4 & 8 \\end{array} \right] \]

Step 3: Second Scalar Multiplication

Multiply each element in matrix \(A\) by 3. \[ 3A = \left[ \begin{array}{ccc}3 \times (-1) & 3 \times 3 & 3 \times 0 \3 \times 2 & 3 \times 4 & 3 \times 1 \\end{array} \right] = \left[ \begin{array}{ccc}-3 & 9 & 0 \6 & 12 & 3 \\end{array} \right] \]

Step 4: Subtraction of Matrices

Subtract the matrix \(3A\) from \(2B\). This is done element-wise.\[ 2B - 3A = \left[ \begin{array}{ccc}0 - (-3) & 4 - 9 & 2 - 0 \6 - 6 & -4 - 12 & 8 - 3 \\end{array} \right] = \left[ \begin{array}{ccc}3 & -5 & 2 \0 & -16 & 5 \\end{array} \right] \]

Step 5: Conclusion

The resulting matrix from the operation \(2B - 3A\) is: \[ \left[ \begin{array}{ccc}3 & -5 & 2 \0 & -16 & 5 \\end{array} \right] \]

Key Concepts

Scalar Multiplication

Scalar multiplication in matrix operations is a fundamental concept that involves multiplying each element within a matrix by a constant value, known as a scalar. Picture scaling a drawing; each line becomes proportionally longer or shorter depending on the scaling factor.
In more formal terms:

• Given a matrix \( M \), a scalar multiplication by a constant \( k \) means that every element \( m_{ij} \) in \( M \) is multiplied by \( k \).
• The resultant matrix has each element as \( k \times m_{ij} \).

Let's consider our example with matrices \( A \) and \( B \). If we multiply matrix \( B \) by 2, then each element of \( B \) gets doubled. Likewise, multiplying \( A \) by 3 means every number in \( A \) becomes three times larger. This operation is straightforward and lays the groundwork for more complex operations like matrix addition or subtraction.

Matrix Subtraction

Matrix subtraction is performed when you have two matrices of the same size and wish to create a new matrix representing the difference between the two. It's akin to subtracting numbers, but here you subtract one matrix from another, element by element.
A few important rules:

• The matrices must be of the same dimensions. For instance, both being 2x3 matrices.
• The subtraction is done on each corresponding element from both matrices.

In the exercise, after achieving \(2B\) and \(3A\) through scalar multiplication, the expression \(2B - 3A\) required subtracting each corresponding element from these two resulting matrices.For example, the top left element of the resulting matrix is arrived at by subtracting the top left element of \(3A\) from \(2B\). Through this step-by-step process, a new matrix is formed clearly showing the difference between these two matrices.

2x3 Matrices

A 2x3 matrix is a rectangular arrangement of numbers with two rows and three columns. It represents a common format for many practical and theoretical calculations within the realm of linear algebra. It's seen frequently in systems of equations, data representation, and transformations.
Understanding the Layout:

• The notation 2x3 describes the structure of the matrix – 2 rows and 3 columns.
• Each element of a matrix is usually denoted in a manner where the row number comes before the column number, like \( a_{ij} \), where \( i \) is the row, and \( j \) is the column.

Such matrices are pivotal in operations like scalar multiplication and subtraction discussed earlier. In these operations, maintaining consistency in the structure while performing calculations is crucial, ensuring that all the operations are mathematically valid.