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Matrix calculations are fundamental in various fields, including mathematics, physics, computer science, and engineering. Here's an overview of common matrix operations and how to perform them:### 1. Matrix AdditionTo add two matrices, they must have the same dimensions. The sum of two matrices \(A\) and \(B\) is a matrix \(C\) where each element \(C_{ij}\) is the sum of the elements \(A_{ij}\) and \(B_{ij}\).\[ C = A + B \]Example:\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \]\[ C = \begin{pmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix} \]### 2. Matrix SubtractionSimilar to addition, to subtract two matrices \(A\) and \(B\), they must have the same dimensions. The difference is given by:\[ C = A - B \]Example:\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \]\[ C = \begin{pmatrix} 1-5 & 2-6 \\ 3-7 & 4-8 \end{pmatrix} = \begin{pmatrix} -4 & -4 \\ -4 & -4 \end{pmatrix} \]### 3. Scalar MultiplicationTo multiply a matrix \(A\) by a scalar \(k\), multiply each element of \(A\) by \(k\).\[ C = kA \]Example:\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad k = 3 \]\[ C = 3 \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 3 \times 1 & 3 \times 2 \\ 3 \times 3 & 3 \times 4 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 9 & 12 \end{pmatrix} \]### 4. Matrix MultiplicationTo multiply two matrices \(A\) and \(B\), the number of columns in \(A\) must be equal to the number of rows in \(B\). The product \(C = AB\) is calculated as follows:\[ C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj} \]Example:\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \]\[ C = AB = \begin{pmatrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \\ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix} \]### 5. Transpose of a MatrixThe transpose of a matrix \(A\) is a new matrix \(A^T\) where the rows of \(A\) become columns and vice versa.Example:\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \]\[ A^T = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix} \]### 6. Determinant of a MatrixFor a square matrix \(A\), the determinant is a scalar value that can be computed recursively or using specific formulas for small matrices.For a 2x2 matrix:\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]\[ \text{det}(A) = ad - bc \]Example:\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \]\[ \text{det}(A) = 1 \cdot 4 - 2 \cdot 3 = 4 - 6 = -2 \]### 7. Inverse of a MatrixThe inverse of a matrix \(A\), denoted \(A^{-1}\), exists if and only if \(A\) is square and \(\text{det}(A) \neq 0\). For a 2x2 matrix:\[ A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]Example:\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \]\[ \text{det}(A) = -2 \]\[ A^{-1} = \frac{1}{-2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} = \begin{pmatrix} -2 & 1 \\ 1.5 & -0.5 \end{pmatrix} \]These are some of the basic operations you can perform with matrices. More advanced operations include eigenvalues and eigenvectors, matrix decompositions, and matrix calculus, which are important in higher-level mathematics and various applications.