Question

a) Gegeben sei \(A=\left(\begin{array}{rr}1 & 2 \\ 4 & -3 \\ 3 & 0\end{array}\right)\) Berechnen Sie a) \(A^{T}\) und b) \(\left(A^{T}\right)^{T}\) b) Gegeben sei \(A=\left(\begin{array}{rrr}1 & 0 & 3 \\ 2 & -3 & 1 \\ 1 & 2 & 2\end{array}\right)\) und \(A^{-1}=\left(\begin{array}{rrr}-8 & 6 & 9 \\ -3 & -1 & 5 \\ 7 & -2 & -3\end{array}\right) \cdot \frac{1}{13}\) Zeigen Sie, daB \(A \cdot A^{-1}=E\).

Short answer

The transpose of matrix \(A = \begin{pmatrix} 1 & 2 \ 4 & -3 \ 3 & 0 \end{pmatrix}\) is \(A^{T} = \begin{pmatrix} 1 & 4 & 3 \ 2 & -3 & 0 \end{pmatrix}\) and \((A^{T})^{T}\) results in the original matrix A. Multiplying the matrix A with its inverse gives the identity matrix E.

Step-by-step solution

Part a: Calculate transpose of Matrix A

The transpose of a matrix is obtained by swapping its rows for columns. Therefore, we calculate it as follows: \(A = \begin{pmatrix} 1 & 2 \ 4 & -3 \ 3 & 0 \end{pmatrix}\) to \(A^{T} = \begin{pmatrix} 1 & 4 & 3 \ 2 & -3 & 0 \end{pmatrix}\).

Part a: Calculate double transpose of A

The double transpose would simply result in the original matrix. That is, \((A^{T})^{T} = A\), therefore \(\left(A^{T}\right)^{T} = \begin{pmatrix} 1 & 2 \ 4 & -3 \ 3 & 0 \end{pmatrix}\) which is the same as our original matrix A.

Part b: Multiply matrix A with its inverse

Matrix multiplication has been given by the equation \(A \cdot A^{-1} = E\) with \(A = \begin{pmatrix} 1 & 0 & 3 \ 2 & -3 & 1 \ 1 & 2 & 2 \end{pmatrix}\) and \(A^{-1} = \begin{pmatrix}-8 & 6 & 9 \ -3 & -1 & 5 \ 7 & -2 & -3\end{pmatrix} \cdot \frac{1}{13}\). This operation has to be carried out position by position for all elements of the two matrices. After performing this operation, you will end up with the identity matrix E, proving that \(A \cdot A^{-1} = E\).

Key concepts

Matrix Inverses

Matrix inverses play a crucial role in solving systems of equations and are akin to division for matrices. When you multiply a matrix by its inverse, the result is the identity matrix, much like multiplying a number by its reciprocal yields one. For a matrix to have an inverse, it must be square (same number of rows and columns) and its determinant should not be zero.
A practical example is found in part b of the exercise, where given a matrix \(A\) and its inverse \(A^{-1}\), multiplying these should result in the identity matrix \(E\). This relationship is mathematically represented as:

• \(A \cdot A^{-1} = E\)

This property is used in numerous applications, such as solving \(AX = B\) for \(X\) by multiplying both sides by \(A^{-1}\) to isolate \(X\). Understanding matrix inverses is not just about learning a mathematical operation; it is about knowing how to apply matrix theories to real-world problems.
When verifying this, check each element within the product matrix to ensure they correspond to the identity matrix's expected results, confirming the inverse matrix's correctness.

Matrix Multiplication

Matrix multiplication is the process of multiplying two matrices by each other. It involves taking the dot product of rows and columns to produce a new matrix. This operation is not commutative; that is, \(A \cdot B eq B \cdot A\) in general. Here's how it works step-by-step:

• Take each row in the first matrix and multiply them with each column in the second matrix.
• For each pair, compute the sum of the products, which becomes an element in the resulting matrix.

In the provided exercise, we saw this in practice as \(A\) was multiplied with its inverse \(A^{-1}\). Matrix multiplication is vital for composition processes in various fields, including computer graphics, physics simulations, and more. When you multiply a matrix \(A\) with its inverse \(A^{-1}\), each pair of corresponding calculations should yield the respective elements of the identity matrix. It’s crucial to track calculations meticulously to ensure accuracy and insight into how matrix interactions work in broader applications. Understanding matrix multiplication is a key step to mastering more complex matrix operations.

Identity Matrix

The identity matrix is like the number 1 for matrix operations; anything multiplied by it remains unchanged. This matrix has ones on the main diagonal (from top left to bottom right) and zeros elsewhere. It's always a square matrix, and its size \(n \times n\) corresponds to the matrices it's associated with.
The exercise highlights the identity matrix through the equation \(A \cdot A^{-1} = E\), where \(E\) signifies the identity matrix of appropriate dimensions. When a matrix \(A\) is multiplied by its inverse \(A^{-1}\), the result should parallel \(E\). This property verifies that the inverse is correctly calculated.

• Diagonal elements are all 1.
• Non-diagonal elements are all 0.
• \(A \cdot I = A\) and \(I \cdot A = A\) for any matrix \(A\) when dimensions align.

Understanding the identity matrix is foundational for comprehending more advanced algebraic structures. It represents an unchanged system even after transformations, embodying the essence of balance and symmetry in matrix operations.