Welche der folgenden 3-rcihigen Matrizen sind orthogonal? $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rrr} 1 & 0 & 1 \\ 2 & -2 & 0 \\ 0 & -1 & 3 \end{array}\right), \quad \mathrm{B}=\frac{1}{3}\left(\begin{array}{rrr} 2 & 2 & 1 \\ 1 & -2 & 2 \\ 2 & -1 & -2 \end{array}\right) \\ &C=\left(\begin{array}{ccc} 0 & 1 / \sqrt{2} & -1 / \sqrt{2} \\ 0 & 1 / \sqrt{2} & 1 / \sqrt{2} \\ 1 & 0 & 0 \end{array}\right) \end{aligned} $$

The transpose of a matrix is an important concept in linear algebra. It is formed by flipping a matrix over its diagonal. This means swapping the row and column indices of the matrix elements. For example, if you have a matrix \[ \textbf{A} = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \ \end{pmatrix} \], its transpose \[ \textbf{A}^T = \begin{pmatrix} a & d & g \ b & e & h \ c & f & i \ \end{pmatrix} \].

The operation of taking the transpose of a matrix is quite simple and useful, especially when working with properties like orthogonality. If \[ \textbf{A} \] is a matrix, then \[ \textbf{A}^T \textbf{A} = \textbf{I} \] holds true if \[ \textbf{A} \] is orthogonal.

The inverse matrix is a crucial concept in linear algebra. For a square matrix \[ \textbf{A} \], its inverse is denoted as \[ \textbf{A}^{-1} \]. If the product of \[ \textbf{A} \textbf{A}^{-1} = \textbf{I} \], where \[ \textbf{I} \] is the identity matrix, then \[ \textbf{A}^{-1} \] is indeed the inverse of \[ \textbf{A} \]. Not all matrices have an inverse; only those that are non-singular (with a non-zero determinant) do.

One important property of orthogonal matrices is that their transpose is also their inverse. This means if \[ \textbf{A} \] is an orthogonal matrix, then \[ \textbf{A}^{-1} = \textbf{A}^T \]. This property makes it easier to find the inverse of orthogonal matrices.

The identity matrix, denoted as \[ \textbf{I} \], is a special kind of square matrix. It has ones on its main diagonal (from top-left to bottom-right) and zeros elsewhere. For example, a 3x3 identity matrix looks like this: \[ \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \ \end{pmatrix} \].

The identity matrix acts as the multiplicative identity in matrix multiplication. This means any matrix \[ \textbf{A} \] multiplied by \[ \textbf{I} \] will give \[ \textbf{A} \] itself, i.e., \[ \textbf{A} \textbf{I} = \textbf{A} \].

In the context of orthogonal matrices, when you multiply an orthogonal matrix by its transpose, you get the identity matrix. Hence, \[ \textbf{A}^T \textbf{A} = \textbf{I} \] signifies that \[ \textbf{A} \] is orthogonal.

Matrix multiplication involves the dot product of rows and columns. For two matrices \[ \textbf{A} \] and \[ \textbf{B} \], the product matrix \[ \textbf{C} \] is obtained by multiplying the elements of the rows of \[ \textbf{A} \] with the elements of the columns of \[ \textbf{B} \] and summing them. This can be formally written as \[ \textbf{C}_{ij} = \textbf{A}_{i1}\textbf{B}_{1j} + \textbf{A}_{i2}\textbf{B}_{2j} + ... + \textbf{A}_{in}\textbf{B}_{nj} \].

Matrix multiplication is not commutative, which means \[ \textbf{A} \textbf{B} eq \textbf{B} \textbf{A} \] in general. However, it is associative and distributive.

In the context of orthogonal matrices, the multiplication of a matrix by its transpose results in the identity matrix. Thus, \[ \textbf{A}^T \textbf{A} = \textbf{I} \] is the key condition to check for orthogonality.