Question

If \(\mathbf{A}=\left(\begin{array}{rr}{1+i} & {-1+2 i} \\ {3+2 i} & {2-i}\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}{i} & {3} \\\ {2} & {-2 i}\end{array}\right),\) find (a) \(\mathbf{A}-2 \mathbf{B}\) (b) \(3 \mathbf{A}+\mathbf{B}\) (c) \(\mathbf{A B}\) (d) BA

Short answer

Based on the given matrices A and B, we have calculated the following operations: a) A - 2B = $\left(\begin{array}{rr}-i & -7+2i \\ -1+2i & 1+3i\end{array}\right)$ b) 3A + B = $\left(\begin{array}{cc}3+4i & -3i \\ 11+6i & 6-i\end{array}\right)$ c) AB = $\left(\begin{array}{cc}5+5i & 15+3i \\ 8+6i & 11+4i\end{array}\right)$ d) BA = $\left(\begin{array}{cc}8+5i & 5+3i \\ 4-4i & -6+6i\end{array}\right)$

Step-by-step solution

Matrices A and B

First, let's write down the given matrices. Matrix A: \(\left(\begin{array}{rr}1+i & -1+2i \\ 3+2i & 2-i\end{array}\right)\) Matrix B: \(\left(\begin{array}{rr}i & 3 \\ 2 & -2i\end{array}\right)\)

A - 2B

Subtract 2 times Matrix B from Matrix A: \(\mathbf{A}-2\mathbf{B}=\left(\begin{array}{rr}1+i & -1+2i \\ 3+2i & 2-i\end{array}\right)-2\left(\begin{array}{rr}i & 3 \\ 2 & -2i\end{array}\right)=\left(\begin{array}{rr}1+i-2i & -1+2i-6 \\ 3+2i-4 & 2-i+4i\end{array}\right)=\left(\begin{array}{rr}-i & -7+2i \\ -1+2i & 1+3i\end{array}\right)\)

3A + B

Add 3 times Matrix A to Matrix B: \(3\mathbf{A}+\mathbf{B}=\left(\begin{array}{cc}3(1+i)+i & 3(-1+2i)+3 \\ 3(3+2i)+2 & 3(2-i)+(-2i)\end{array}\right)=\left(\begin{array}{cc}3+3i+i & -3+6i+3 \\ 9+6i+2 & 6-3i-2i\end{array}\right)=\left(\begin{array}{cc}3+4i & -3i \\ 11+6i & 6-i\end{array}\right)\)

AB

To multiply A and B, perform each element operation like this: \(\mathbf{A}\mathbf{B}=\left(\begin{array}{cc}(1+i)(i)+(3)(-1+2i) & (1+i)(3)+(3)(2) \\ (3+2i)(i)+(2-i)(2) & (3+2i)(3)+(2-i)(-2i)\end{array}\right)=\left(\begin{array}{cc}-1+2i+3i+6 & 3i+9+6 \\ -2i+6i+6+2+2i & 9+6i-2i+2\end{array}\right)=\left(\begin{array}{cc}5+5i & 15+3i \\ 8+6i & 11+4i\end{array}\right)\)

BA

To multiply B and A, perform each element operation like this: \(\mathbf{B}\mathbf{A}=\left(\begin{array}{cc}(i)(1+i)+(3)(3+2i) & (i)(-1+2i)+(3)(2-i) \\ (2)(1+i)+(-2i)(3+2i) & (2)(-1+2i)+(-2i)(2-i)\end{array}\right)=\left(\begin{array}{cc}-1i+1i+9+6i & -1i+2i^2+6-3i \\ 2+2i-6i+2i^2 & -2+4i+4i^2+4+2i\end{array}\right) =\left(\begin{array}{cc}8+5i & 5+3i \\ 4-4i & -6+6i\end{array}\right)\)

Key concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra, involving the element-wise addition of two matrices of the same dimensions. To perform matrix addition, simply add corresponding entries from each matrix together to form a new matrix. It's important to note that you can only add matrices if they're the same size; otherwise, the operation isn't defined.

Consider two matrices, \( \text{Matrix A} \) and \( \text{Matrix B} \), with identical dimensions. The sum \( \text{Matrix C} = \text{Matrix A} + \text{Matrix B} \) is computed by adding each element \( a_{ij} \) of \( \text{Matrix A} \) to the corresponding element \( b_{ij} \) of \( \text{Matrix B} \), resulting in a new matrix where each element \( c_{ij} = a_{ij} + b_{ij} \). This operation can be visually depicted in a step-by-step manner, which assists in avoiding any potential confusion.

Matrix Scalar Multiplication

Matrix scalar multiplication involves multiplying every element of a matrix by a scalar, which is a single number. This process changes the magnitude of the matrix elements but not the matrix's dimensions.

When you multiply a matrix, say \( \text{Matrix A} \), by a scalar \( k \) to get \( \text{Matrix B} = k \times \text{Matrix A} \), each element \( b_{ij} = k \times a_{ij} \) in the resulting matrix is the product of the scalar and the corresponding element in the original matrix. The scalar can be any real or complex number, and this type of multiplication distributes over matrix addition. Pictorial representations or step-by-step calculations can make this process easier to understand, enhancing clarity for students.

Matrix Multiplication

Matrix multiplication is slightly more complex than addition or scalar multiplication. In this operation, you multiply two matrices by taking the dot product of the rows of the first matrix with the columns of the second matrix. The number of columns in the first matrix must equal the number of rows in the second matrix for the operation to be valid.

To obtain the element \( c_{ij} \) of the product matrix \( \text{Matrix C} = \text{Matrix A} \times \text{Matrix B} \), you compute the sum of products of the corresponding entries from the \( i \)th row of \( \text{Matrix A} \) and the \( j \)th column of \( \text{Matrix B} \). Demonstrating this using concrete examples, and working through each step in the calculation, ensures better comprehension, as matrix multiplication is not commutative, meaning \( \text{Matrix A} \times \text{Matrix B} \) generally does not equal \( \text{Matrix B} \times \text{Matrix A} \).

Complex Numbers

Complex numbers extend the idea of the one-dimensional number line to a two-dimensional complex plane, using the concept of 'imaginary' units. A complex number has both a real part and an imaginary part and is usually written in the form \( a + bi \), where \( a \) is the real component, \( b \) is the imaginary component, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \).

In matrix operations involving complex numbers, such as those in the given exercise, each element of the matrices \( \text{Matrix A} \) and \( \text{Matrix B} \) is a complex number. The operations of addition, scalar multiplication, and matrix multiplication are performed considering both the real and imaginary parts. Using step-by-step examples that detail the handling of real and imaginary components provides students with a clear method for working with complex matrices.