Problem 14 - Advanced Engineering Mathematics

Question

In Problems 13 and 14, find the entries \(c_{23}\) and \(c_{12}\) for the matrix \(\mathbf{C}=2 \mathbf{A}-3 \mathbf{B}\). $$ \mathbf{A}=\left(\begin{array}{rrr} 1 & -1 & 1 \\ 2 & 2 & 1 \\ 0 & -4 & 1 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{lll} 2 & 0 & 5 \\ 0 & 4 & 0 \\ 3 & 0 & 7 \end{array}\right) $$

Step-by-Step Solution

Answer

\(c_{23} = 2\); \(c_{12} = -2\).

Step 1: Determine Matrix Dimensions

Identify that both matrices \(\mathbf{A}\) and \(\mathbf{B}\) are of size 3x3. Hence, matrix \(\mathbf{C}=2\mathbf{A} - 3\mathbf{B}\) will also be a 3x3 matrix.

Step 2: Calculate Matrix C

Compute matrix \(\mathbf{C} = 2\mathbf{A} - 3\mathbf{B}\) by applying scalar multiplication to \(\mathbf{A}\) and \(\mathbf{B}\), and then perform matrix subtraction.

Step 3: Scalar Multiplication with A

Calculate \(2 \mathbf{A}\) by multiplying each entry of \(\mathbf{A}\) by 2: \[ 2 \mathbf{A} = \begin{pmatrix} 2 & -2 & 2 \ 4 & 4 & 2 \ 0 & -8 & 2 \end{pmatrix} \]

Step 4: Scalar Multiplication with B

Calculate \(3 \mathbf{B}\) by multiplying each entry of \(\mathbf{B}\) by 3: \[ 3 \mathbf{B} = \begin{pmatrix} 6 & 0 & 15 \ 0 & 12 & 0 \ 9 & 0 & 21 \end{pmatrix} \]

Step 5: Matrix Subtraction to Find C

Subtract \(3 \mathbf{B}\) from \(2 \mathbf{A}\) to find \(\mathbf{C}\):\[ \mathbf{C} = 2\mathbf{A} - 3\mathbf{B} = \begin{pmatrix} 2 & -2 & 2 \ 4 & 4 & 2 \ 0 & -8 & 2 \end{pmatrix} - \begin{pmatrix} 6 & 0 & 15 \ 0 & 12 & 0 \ 9 & 0 & 21 \end{pmatrix} = \begin{pmatrix} -4 & -2 & -13 \ 4 & -8 & 2 \ -9 & -8 & -19 \end{pmatrix} \]

Step 6: Extract Entry c23

Identify the entry at position \((2,3)\) of matrix \(\mathbf{C}\), which is \(c_{23} = 2\).

Step 7: Extract Entry c12

Identify the entry at position \((1,2)\) of matrix \(\mathbf{C}\), which is \(c_{12} = -2\).

Key Concepts

Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a scalar (a single number). This operation is straightforward yet fundamental in linear algebra.
For example, to find \(2 \mathbf{A}\), where \(\mathbf{A}\) is a 3x3 matrix, you simply multiply every entry in \(\mathbf{A}\) by 2. Here’s how you do it:

• Multiply the first row: \((1, -1, 1)\) by 2 results in \((2, -2, 2)\).
• Multiply the second row: \((2, 2, 1)\) by 2 results in \((4, 4, 2)\).
• Multiply the third row: \((0, -4, 1)\) by 2 results in \((0, -8, 2)\).

After applying scalar multiplication, matrix \(2\mathbf{A}\) becomes a new matrix with each element twice as large as the corresponding element in \(\mathbf{A}\). This operation is used extensively in solving systems of equations and transformations in geometry.

Matrix Subtraction

Matrix subtraction is an operation where corresponding elements of two matrices of the same dimensions are subtracted from each other.
An important rule to remember now is that matrix subtraction can only be performed on matrices that have the same dimensions.
To subtract one 3x3 matrix, such as \(3\mathbf{B}\), from another 3x3 matrix, \(2\mathbf{A}\), we proceed by subtracting each element in \(3\mathbf{B}\) from the corresponding element in \(2\mathbf{A}\).

• Let's subtract the first row: \[(2, -2, 2) - (6, 0, 15) = (-4, -2, -13)\].
• Next, the second row: \[(4, 4, 2) - (0, 12, 0) = (4, -8, 2)\].
• Finally, the third row: \[(0, -8, 2) - (9, 0, 21) = (-9, -8, -19)\].

Thus, the resultant matrix from this subtraction represents the changes made when each pair of corresponding elements from \(2\mathbf{A}\) and \(3\mathbf{B}\) interact. This is useful in applications involving changes or differences between datasets, like comparing different states within a system.

Matrix Dimensions

Matrix dimensions describe the size of a matrix, given by the number of rows and columns it contains. For example, both matrices \(\mathbf{A}\) and \(\mathbf{B}\) are 3x3 matrices, meaning they each have three rows and three columns.
Knowing the dimensions of a matrix is essential because it dictates how you can perform operations like scalar multiplication, addition, and subtraction.

• Only matrices with the same dimensions can be added or subtracted.
• In scalar multiplication, any matrix can be multiplied by a scalar regardless of its dimensions because the scalar does not change the structure, only the values.

When working with matrices, always start by checking their dimensions to ensure that the intended operations are permissible.
Dimensions also play a critical role in defining other matrix characteristics, such as being invertible or having consistent systems of equations.