Question

If \(\mathbf{A}=\left(\begin{array}{rrr}{1} & {-2} & {0} \\ {3} & {2} & {-1} \\\ {-2} & {1} & {3}\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rrr}{4} & {-2} & {3} \\ {-1} & {5} & {0} \\\ {6} & {1} & {2}\end{array}\right),\) find (a) \(2 \mathrm{A}+\mathrm{B}\) (b) \(\mathbf{A}-4 \mathbf{B}\) (c) \(\mathrm{AB}\) (d) BA

Short answer

Answer: The result of the matrix multiplication AB is: \(\mathbf{AB}=\left(\begin{array}{rrr}{6} & {-10} & {3} \\\ {4} & {5} & {7} \\\ {14} & {3} & {3}\end{array}\right)\)

Step-by-step solution

Perform Scalar Multiplication of A

Multiply each element in matrix A by 2. \(2\mathbf{A} = \left(\begin{array}{rrr}{2} & {-4} & {0} \\\ {6} & {4} & {-2} \\\ {-4} & {2} & {6}\end{array}\right)\)

Find 2A + B

Add the corresponding elements of matrices 2A and B. \(2\mathbf{A} + \mathbf{B} = \left(\begin{array}{rrr}{2+4} & {-4-2} & {0+3} \\\ {6-1} & {4+5} & {-2+0} \\\ {-4+6} & {2+1} & {6+2}\end{array}\right) = \left(\begin{array}{rrr}{6} & {-6} & {3} \\\ {5} & {9} & {-2} \\\ {2} & {3} & {8}\end{array}\right)\)

Perform Scalar Multiplication of B

Multiply each element in matrix B by 4. \(4\mathbf{B} = \left(\begin{array}{rrr}{16} & {-8} & {12} \\\ {-4} & {20} & {0} \\\ {24} & {4} & {8}\end{array}\right)\)

Find A - 4B

Subtract the corresponding elements of matrices 4B from A. \(\mathbf{A} - 4\mathbf{B} = \left(\begin{array}{rrr}{1-16} & {-2 - (-8)} & {0-12} \\\ {3 - (-4)} & {2-20} & {-1-0} \\\ {-2-24} & {1-4} & {3-8}\end{array}\right) = \left(\begin{array}{rrr}{-15} & {6} & {-12} \\\ {7} & {-18} & {-1} \\\ {-26} & {-3} & {-5}\end{array}\right)\)

Calculate AB (Matrix Multiplication)

Multiply matrix A and B as per matrix multiplication rule. To calculate the element \(\mathrm{c}_{ij}\) in matrix \(\mathbf{AB}\), multiply the \(i^{th}\) row of matrix \(\mathbf{A}\) by the \(j^{th}\) column of matrix \(\mathbf{B}\). $\mathbf{AB} = \left(\begin{array}{rrr} {(1)(4)+(-2)(-1)+(0)(6)} & {(1)(-2)+(-2)(5)+(0)(1)} & {(1)(3)+(-2)(0)+(0)(2)} \\ {(3)(4)+(2)(-1)+(-1)(6)} & {(3)(-2)+(2)(5)+(-1)(1)} & {(3)(3)+(2)(0)+(-1)(2)} \\ {(-2)(4)+(1)(-1)+(3)(6)} & {(-2)(-2)+(1)(5)+(3)(1)} & {(-2)(3)+(1)(0)+(3)(2)}\\ \end{array}\right) = \left(\begin{array}{rrr}{6} & {-10} & {3} \\\ {4} & {5} & {7} \\\ {14} & {3} & {3}\end{array}\right)$

Calculate BA (Matrix Multiplication)

Multiply matrix B and A as per matrix multiplication rule. To calculate the element \(\mathrm{d}_{ij}\) in matrix \(\mathbf{BA}\), multiply the \(i^{th}\) row of matrix \(\mathbf{B}\) by the \(j^{th}\) column of matrix \(\mathbf{A}\). $\mathbf{BA} = \left(\begin{array}{rrr} {(4)(1)+(-2)(3)+(-3)(-2)} & {(4)(-2)+(-2)(2)+(3)(1)} & {(4)(0)+(-2)(-1)+(3)(3)} \\ {(-1)(1)+(5)(3)+(0)(-2)} & {(-1)(-2)+(5)(2)+(0)(1)} & {(-1)(0)+(5)(-1)+(0)(3)} \\ {(6)(1)+(1)(3)+(-2)(2)} & {(6)(-2)+(1)(2)+(2)(1)} & {(6)(0)+(1)(-1)+(2)(3)}\\ \end{array}\right) = \left(\begin{array}{rrr}{2} & {-6} & {11} \\\ {14} & {8} & {-5} \\\ {7} & {-9} & {5}\end{array}\right)$ The final results are: (a) \(2\mathbf{A}+\mathbf{B}=\left(\begin{array}{rrr}{6} & {-6} & {3} \\\ {5} & {9} & {-2} \\\ {2} & {3} & {8}\end{array}\right)\) (b) \(\mathbf{A}-4\mathbf{B}=\left(\begin{array}{rrr}{-15} & {6} & {-12} \\\ {7} & {-18} & {-1} \\\ {-26} & {-3} & {-5}\end{array}\right)\) (c) \(\mathbf{AB}=\left(\begin{array}{rrr}{6} & {-10} & {3} \\\ {4} & {5} & {7} \\\ {14} & {3} & {3}\end{array}\right)\) (d) \(\mathbf{BA}=\left(\begin{array}{rrr}{2} & {-6} & {11} \\\ {14} & {8} & {-5} \\\ {7} & {-9} & {5}\end{array}\right)\)

Key concepts

Scalar Multiplication

Scalar multiplication is the process of multiplying each element of a matrix by a scalar, which is a single number. It's like applying the multiplication operation individually to every element within the matrix.
This operation scales the entire matrix without changing its dimensions. It's a straightforward way to increase or decrease the values in a matrix proportionally.
For example, given a matrix \( \mathbf{A} = \begin{pmatrix} 1 & -2 & 0 \ 3 & 2 & -1 \ -2 & 1 & 3 \end{pmatrix} \) and a scalar \( 2 \), multiplying the matrix by 2 means:
* Multiply each entry in \( \mathbf{A} \) by 2.
* Resulting in another matrix, \( 2\mathbf{A} = \begin{pmatrix} 2 & -4 & 0 \ 6 & 4 & -2 \ -4 & 2 & 6 \end{pmatrix} \).
This operation is helpful when you need to adjust the strength of connections in a network or scale transformations in graphics.

Matrix Addition and Subtraction

Matrix addition and subtraction involve adding or subtracting corresponding elements from two matrices. This operations require the matrices to have the same dimensions.
Simply put, each element in one matrix is paired with the corresponding element in the other matrix.
Here's how it works for addition:

• Take two matrices \( \mathbf{A} \) and \( \mathbf{B} \), where both are of the same size.
• Add elements located at the same position:

\( 2\mathbf{A} + \mathbf{B} = \begin{pmatrix} 2+4 & -4+(-2) & 0+3 \ 6-1 & 4+5 & -2+0 \ -4+6 & 2+1 & 6+2 \end{pmatrix} = \begin{pmatrix} 6 & -6 & 3 \ 5 & 9 & -2 \ 2 & 3 & 8 \end{pmatrix} \).
Matrix subtraction follows a similar line of action, except you subtract the corresponding elements.

• For example, if you subtract matrix \( 4\mathbf{B} \) from \( \mathbf{A} \), you would calculate:
• \( \mathbf{A} - 4\mathbf{B} = \begin{pmatrix} 1-16 & -2-(-8) & 0-12 \ 3-(-4) & 2-20 & -1-0 \ -2-24 & 1-4 & 3-8 \end{pmatrix} = \begin{pmatrix} -15 & 6 & -12 \ 7 & -18 & -1 \ -26 & -3 & -5 \end{pmatrix} \).

Addition and subtraction of matrices is very useful in areas such as economics for calculating net balances, or physics for analyzing forces.

Matrix Multiplication

Matrix multiplication is slightly more complex than addition or subtraction, and involves multiplying rows by columns.
For two matrices to multiply, the number of columns in the first must match the number of rows in the second.
This operation results in a new matrix with dimensions taken from the rows of the first and columns of the second matrix.
Here is how it is done:

• Each element of the resulting matrix is calculated by taking a row from the first matrix and a column from the second matrix.
• Then, perform a dot product: multiply corresponding elements and sum them up.

For matrix \( \mathbf{AB} \), where \( \mathbf{A} \) is a 3x3 matrix and \( \mathbf{B} \) is another 3x3 matrix:

• The element in the first row, first column is \((1 \cdot 4) + (-2 \cdot -1) + (0 \cdot 6) = 6\).
• Continue this process for each element in the resulting matrix.

This results in \( \mathbf{AB} = \begin{pmatrix} 6 & -10 & 3 \ 4 & 5 & 7 \ 14 & 3 & 3 \end{pmatrix} \).
Matrix multiplication is a cornerstone of linear algebra, extensively used in fields like engineering for systems modeling and computer graphics for transformations.