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# Matrices ## Matrix definition * A matrix is a rectangular array of numbers or symbols. * Matrices are used to represent linear transformations, solve systems of linear equations, and study vector spaces. * The dimensions of a matrix are given by the number of rows and columns it has....

# Matrices ## Matrix definition * A matrix is a rectangular array of numbers or symbols. * Matrices are used to represent linear transformations, solve systems of linear equations, and study vector spaces. * The dimensions of a matrix are given by the number of rows and columns it has. ## Matrix notation * Matrices are usually denoted by uppercase letters, such as A, B, or C. * The elements of a matrix are usually denoted by lowercase letters with subscripts indicating their row and column position, such as $a_{ij}$ or $b_{ij}$. * For example, if A is a matrix with m rows and n columns, then we write $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$ ## Types of matrices * **Square matrix**: A matrix with the same number of rows and columns. * **Diagonal matrix:** A square matrix in which all the elements outside the main diagonal are zero. * **Identity matrix:** A diagonal matrix in which all the elements on the main diagonal are one. * **Zero matrix:** A matrix in which all the elements are zero. * **Transpose matrix:** The transpose of a matrix A, denoted by $A^T$, is obtained by interchanging the rows and columns of A. ## Matrix operations * **Addition and subtraction:** Matrices can be added or subtracted if they have the same dimensions. The sum or difference of two matrices is obtained by adding or subtracting the corresponding elements. * **Scalar multiplication:** A matrix can be multiplied by a scalar by multiplying each element of the matrix by the scalar. * **Matrix multiplication:** The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B. The element in the i-th row and j-th column of the product AB is obtained by taking the dot product of the i-th row of A and the j-th column of B. ## Example A = $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, B = $\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$ $A + B = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$ $A \times B = \begin{bmatrix} 1\times5 + 2\times7 & 1\times6 + 2\times8 \\ 3\times5 + 4\times7 & 3\times6 + 4\times8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$