Problem 34 - Advanced Engineering Mathematics

Question

$$, Suppose \(\mathbf{A}=\left(\begin{array}{rr}2 & 4 \\ -3 & 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}4 & 10 \\ 2 & 5\end{array}\right)\) Verify the given property by computing the left and right members of the given equality. $$ (6 \mathbf{A})^{T}=6 \mathbf{A}^{T} $$

Step-by-Step Solution

Answer

The equality \((6\mathbf{A})^T = 6\mathbf{A}^T\) holds true.

Step 1: Calculate 6A

To start, multiply each element of matrix \(\mathbf{A}\) by 6.\[6\mathbf{A} = 6 \times \left(\begin{array}{rr}2 & 4 \ -3 & 2\end{array}\right) = \left(\begin{array}{rr}12 & 24 \ -18 & 12\end{array}\right)\]

Step 2: Transpose 6A

Now, find the transpose of \(6\mathbf{A}\) by swapping its rows and columns.\[(6\mathbf{A})^T = \left(\begin{array}{rr}12 & -18 \ 24 & 12\end{array}\right)\]

Step 3: Calculate AT

Next, find the transpose of the matrix \(\mathbf{A}\).\[\mathbf{A}^T = \left(\begin{array}{rr}2 & -3 \ 4 & 2\end{array}\right)\]

Step 4: Multiply AT by 6

After finding \(\mathbf{A}^T\), multiply each element of this transpose by 6.\[6\mathbf{A}^T = 6 \times \left(\begin{array}{rr}2 & -3 \ 4 & 2\end{array}\right) = \left(\begin{array}{rr}12 & -18 \ 24 & 12\end{array}\right)\]

Step 5: Compare Results

Finally, compare the results of \((6\mathbf{A})^T\) and \(6\mathbf{A}^T\). Both matrices are:\[\left(\begin{array}{rr}12 & -18 \ 24 & 12\end{array}\right)\]Thus, the given equality \((6\mathbf{A})^T = 6\mathbf{A}^T\) holds true.

Key Concepts

Matrix Transposition

Matrix transposition is an essential operation for understanding many algebraic concepts. To transpose a matrix, swap its rows and columns. This means that the first row becomes the first column, the second row becomes the second column, and so on. If you start with a matrix like: \[ \begin{bmatrix}a & b \ c & d \end{bmatrix} \] The transpose of this matrix would be: \[ \begin{bmatrix}a & c \ b & d \end{bmatrix} \]
Transpositions help change the orientation of a matrix without affecting its elements' values. This operation is handy in various scenarios, such as solving linear equations or transforming data in different directions. Always remember that transposing doesn't alter the size of the matrix; a \(2 \times 3\) matrix will become a \(3 \times 2\) matrix after transposition.
One meaningful property of transposing is that when you transpose a transposed matrix, you get back the original matrix. Essentially,

• \((\mathbf{A}^T)^T = \mathbf{A}\)

This property is what comes into play when verifying computations involving transpositions.

Scalar Multiplication

Scalar multiplication in matrix operations involves multiplying each element of a matrix by a single number, known as a scalar. This operation is straightforward and constitutes one of the basic matrix manipulations. For example, if you start with: \[ \mathbf{A} = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \] and you multiply by a scalar \(k = 3\): \[ 3\mathbf{A} = \begin{bmatrix} 3 \times 1 & 3 \times 2 \ 3 \times 3 & 3 \times 4 \end{bmatrix} = \begin{bmatrix} 3 & 6 \ 9 & 12 \end{bmatrix} \]
When you multiply by a scalar, the operation spreads uniformly across the matrix. This operation does not change the structural layout of the matrix; instead, it adjusts the magnitude of each element. Scalar multiplication is useful in various practical applications, such as when scaling datasets or adjusting parameters in equations. Always notice that the multiplication is element-wise; thus, each component of the matrix scales identically.
Lastly, it is essential to know that when a scalar multiplies a matrix, if the scalar is zero, the result is a zero matrix, maintaining the same size as the original.

Matrix Equality

Matrix equality is a concept where two matrices are considered equal if they have the same dimensions and their corresponding elements are identical. So, for matrices \(\mathbf{A}\) and \(\mathbf{B}\) to be equal, they must satisfy the conditions:

• \(\mathbf{A}_{ij} = \mathbf{B}_{ij}\) for all \(i\) and \(j\)
• \(\mathbf{A}\) and \(\mathbf{B}\) have the same number of rows and columns

For example, consider matrices: \[ \mathbf{A} = \begin{bmatrix}1 & 2 \ 3 & 4\end{bmatrix} \] and \[ \mathbf{B} = \begin{bmatrix}1 & 2 \ 3 & 4\end{bmatrix} \] Since every element corresponds equally and both are \(2 \times 2\) matrices, \(\mathbf{A} = \mathbf{B}\).
Matrix equality comes into play in many mathematical proofs and problem-solving scenarios, such as finding solutions to equations involving matrices or checking results for matrix operations' correctness. Understanding matrix equality ensures a strong foundation for more advanced topics in linear algebra and helps greatly in ensuring accuracy when dealing with matrices. Understanding how to compare matrices is fundamental for effective data analysis, solving linear systems, and implementing algorithms that require matrix manipulations.