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ss_10_5

# ss_10_5 - Section 10.5 Matrix Algebra In this section we...

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Section 10.5 Matrix Algebra In this section, we define matrix operations such as addition, subtraction, and multiplication, and we give a procedure for finding the inverse of intertible matrices. A matrix is a rectangular array of numbers. A matrix that has m rows and n columns is said to be an m × n matrix, and the m × n matrix is said to be of order m × n . The element a ij is in row i and column j of the matrix ( a ij ). ( a ij ) = a 11 a 12 · · · a 1 j · · · a 1 n a 21 a 22 · · · a 2 j · · · a 2 n · · · a i 1 a i 2 · · · a ij · · · a in · · · a m 1 a m 2 · · · a mj · · · a mn Examples of matrices. 1 2 - 1 3 - 2 2 1 2 - 1 3 2 4 1 - 1 0 2 1 2 3 4 17 Matrix Equality. Two matrices A = ( a ij ) and B = ( b ij ) are equal iff a ij = b ij for all i, j . Note: Two matrices are equal iff corresponding entries are identical. Example. 1 2 3 4 = 1 2 3 4 2 1 3 4 6 = 1 2 3 4 Adding and Subtracting Matrices Only matrices of the same order can be added and subtracted. Matrices are added and sub- tracted term by term, i.e., if A = ( a ij ) and B = ( b ij ), then A + B = ( a ij + b ij ) and A - B = ( a ij - b ij ). Example. If A = 1 1 - 2 3 and B = 2 0 - 4 7 , then A + B = 3 1 - 6 10 A - B = - 1 1 2 - 4 Let α be a scalar and let A = ( a ij ) be a matrix. Then scalar multiplication is defined by αA = ( αa ij ). Example. Let α = 2 and A = 1 2 - 2 3 . Then αA = 2 1 2 - 2 3 = 2 4 - 4 6 Some Properties of matrix addition 1

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A + B = B + A A + ( B + C ) = ( A + B ) + C ( k + h ) A = kA + hA k ( A + B ) = kA + kB Multiplication of Matrices The 1 × n matrix R = [ r 1 , r 2 , · · · , r n ] is called a row vector . The m × 1 matrix C = c 1 c 2 .
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