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Contents Adding and Subtracting Matrices Scalar Multiplication Matrix by Matrix Multiplication Transpose of a Matrix Powers of a Matrix Determinant of a Matrix Inverse of Matrix Adding and Subtracting Matrices Matrices can only be added or subtracted if they have the same dimension. Addition or subtraction is achieved by adding or subtracting corresponding elements. Example A = \left(\begin{matrix} 8 & 5 \\ 6 & 7 \end{matrix}\right) B = \left(\begin{matrix} 3 & 1 \\ 7 & 2 \end{matrix}\right) A + B = \left(\begin{matrix} 8+3 & 5+1 \\ 6+7 & 7+2 \end{matrix}\right) = \left(\begin{matrix} 11 & 6 \\ 13 & 9 \end{matrix}\right) A - B = \left(\begin{matrix} 8-3 & 5-1 \\ 6-7 & 7-2 \end{matrix}\right) = \left(\begin{matrix} 5 & 4 \\ -1 & 5 \end{matrix}\right) Scalar Multiplication Every element of the matrix is multiplied by a value. Example A = \left(\begin{matrix} 8 & 5 \\ 6 & 7 \end{matrix}\right) 4A = 4 \left(\begin{matrix} 8 & 5 \\ 6 & 7 \end{matrix}\right) = \left(\begin{matrix} 4 \cdot 8 & 4 \cdot 5 \\ 4 \cdot 6 & 4 \cdot 7 \end{matrix}\right) = \left(\begin{matrix} 32 & 20 \\ 24 & 28 \end{matrix}\right) Matrix by Matrix Multiplication Given matrices A and B. You can multiply A and B only if the number of columns of matrix A are equal to the number of rows of matrix B. Multiply each value of each row of matrix A with corresponding values in the columns of matrix B. \left(\begin{matrix} a & b & c \\ d & e & f \end{matrix}\right) \left(\begin{matrix} u & v \\ w & x \\ y & z \end{matrix}\right) = \left(\begin{matrix} au+bw+cy & av+bx+cz \\ du+ew+fy & dv+ex+fz \end{matrix}\right) Example AB = \left(\begin{matrix} 2 & -3 & 9 \\ 5 & 4 & 7 \end{matrix}\right) \left(\begin{matrix} 8 & 10 \\ 2 & 3 \\ 1 & 2 \end{matrix}\right) = \left(\begin{matrix} 2 \cdot 8 + -3 \cdot 2 + 9 \cdot 1 & 2 \cdot 10 + -3 \cdot 3 + 9 \cdot 2 \\ 5 \cdot 8 + 4 \cdot 2 + 7 \cdot 1 & 5 \cdot 10 + 4 \cdot 3 + 7 \cdot 2 \end{matrix}\right) = \left(\begin{matrix} 19 & 29 \\ 55 & 76 \end{matrix}\right) Please Note AB \ne BA Example \left(\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}\right) \left(\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}\right) = \left(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}\right) and \left(\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}\right) \left(\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}\right) = \left(\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}\right) Transpose of a Matrix A^{T}, the transpose of matrix A is a matrix which is formed by interchanging rows and columns of matrix A. Example A = \left(\begin{matrix} 5 & 3 & 9 \\ 2 & 1 & 8 \end{matrix}\right) A is a 2×3 matrix. A^{T} is a 2×3 matrix. Interchange rows and columns to get the transpose. A^{T} = \left(\begin{matrix} 5 & 2 \\ 3 & 1 \\ 9 & 8 \end{matrix}\right) Powers of a Matrix A^{n}, is defined as n copies of matrix A. Example A = \left(\begin{matrix} 5 & 2 \\ 3 & 1 \end{matrix}\right) A^{2} = \left(\begin{matrix} 5 & 2 \\ 3 & 1 \end{matrix}\right) \left(\begin{matrix} 5 & 2 \\ 3 & 1 \end{matrix}\right) A^{3} = \left(\begin{matrix} 5 & 2 \\ 3 & 1 \end{matrix}\right) \left(\begin{matrix} 5 & 2 \\ 3 & 1 \end{matrix}\right) \left(\begin{matrix} 5 & 2 \\ 3 & 1 \end{matrix}\right) A^{4} = \left(\begin{matrix} 5 & 2 \\ 3 & 1 \end{matrix}\right) \left(\begin{matrix} 5 & 2 \\ 3 & 1 \end{matrix}\right) \left(\begin{matrix} 5 & 2 \\ 3 & 1 \end{matrix}\right) \left(\begin{matrix} 5 & 2 \\ 3 & 1 \end{matrix}\right) Determinant of a Matrix |A| is a used to symbolize the determinant of matrix A. Only a square matrix can have a determinant. Finding the determinant 2×2 Matrix A = \left(\begin{matrix} a & b \\ c & d \end{matrix}\right) |A| = ad-bc Example A = \left(\begin{matrix} 5 & 3 \\ 2 & 8 \end{matrix}\right) |A| = (5)(8)-(3)(2) = 34 3×3 Matrix A = \left(\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}\right) |A| = a(ei - fh) - b(di-fg) + c(dh-eg) The image below shows how this formula is calculated. Example A = \left(\begin{matrix} 5 & 3 & 4 \\ 7 & 2 & 6 \\ 2 & 1 & 9 \end{matrix}\right) |A| = 5((2)(9)-(6)(1)) - 3((7)(9)-(6)(2)) + 4((7)(1)-(2)(2)) = -81 4×4 Matrix A = \left(\begin{matrix} a & b & c & d \\ e & f & g & h \\ i & j & k & l \\ m & n & o & p \end{matrix}\right) |A| = a \left|\begin{matrix} f & g & h \\ j & k & l \\ n & o & p \end{matrix}\right| - b \left|\begin{matrix} e & g & h \\ i & k & l \\ m & o & p \end{matrix}\right| + c \left|\begin{matrix} e & f & h \\ i & j & l \\ m & n & p \end{matrix}\right| - d \left|\begin{matrix} e & f & g \\ i & j & k \\ m & n & o \end{matrix}\right| Inverse of a Matrix The inverse of a matrix A is a matrix A^{-1} such that AA^{-1}=I . I is the identity matrix. Please Note 1. Only square matrices can have an inverse. 2. If the determinant of a matrix is 0, that matrix has no inverse. Finding the Inverse of a Matrix One method of finding the inverse is to use Gauss-Jordan elimination to transform [A|I] to [I|A^{-1}] Example A = \left(\begin{matrix} 8 & 5 \\ 6 & 4 \end{matrix}\right) Find inverse of A. \left[ \begin{array}{cc|cc} 8 & 5 & 1 & 0 \\ 6 & 4 & 0 & 1 \end{array} \right] R_{1} \rightarrow \frac{1}{8}R_{1} \left[ \begin{array}{cc|cc} 1 & \frac{5}{8} & \frac{1}{8} & 0 \\ 6 & 4 & 0 & 1 \end{array} \right] R_{2} \rightarrow R_{2}- 6R_{1} \left[ \begin{array}{cc|cc} 1 & \frac{5}{8}

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Encoding matrix (F×K), sub-matrix of T B encoding matrix (F×F), sub-matrix of T B−1 solution matrix (F×F) C encoded data matrix ((N×M)×L)= ( ( N + M ) × L ) = [ D J ] C′ surviving encoded data matrix ( N × L ) = [ X W ] D original data matrix (N×L) D′ permuted original data matrix ( N × L ) = [ X Y ] E information dispersal matrix (IDM) ( ( N + M ) × N ) = [ I N H ] F number of failed data drives G number of failed check drives H check drive encoding matrix (M×N) I identity matrix (JK=K×K identity matrix, IN=N×N identity matrix) J encoded check data matrix (M×L) K number of surviving data drives=N−F L data block size (elements or bytes) M number of check drives Mmax maximum value of M N number of data drives Nmax maximum value of N O zero matrix (K×F), sub-matrix of T S master encoding matrix ((Mmax+Nmax)×Nmax) T transformed IDM ( N × N ) = [ I K O A B ] W surviving check data matrix (F×L) X surviving original data matrix (K×L) Y lost original data matrix (F×L) DETAILED DESCRIPTION (NEW) While optimal erasure codes have many applications, for ease of description, they will be described in this application primarily with respect to RAID applications, i.e., erasure coding systems for the storage and retrieval of digital data distributed across numerous storage devices (or drives), though. Browse 5,174 amazing Matrix stock footage videos for royalty-free download from the creative contributors at Vecteezy! matrix code; matrix background; zoom matrix; matrix looping; animated matrix; matrix numbers; red matrix; digital matrix; data matrix; matrix effect; animation matrix; matrix blue; white matrix; 4k matrix; matrix mp4

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