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Matrices

Matrices. A matrix is a rectangular array of quantities (numbers, expressions or function), arranged in m rows and n columns. 3 1 1 -2 -2 3 0. 2x 3y 4z. 5 -2 1. a11 a12 a13 a21 a22 a23 a31 a32 a33. 0 0 0 1 0 0 0 1. Special Matrices. Identity matrix. m = n Square matrix.

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Matrices

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  1. Matrices • A matrix is a rectangular array of quantities (numbers, expressions or function), arranged in m rows and n columns. • 3 1 • 1 -2 • -2 3 0 2x 3y 4z 5 -2 1

  2. a11 a12 a13 a21 a22 a23 a31 a32 a33 • 0 0 • 0 1 0 • 0 0 1 Special Matrices Identity matrix m = n Square matrix a11 0 0 0 a22 0 0 0 a33 Column matrix a11 a12 a13 A11 A21 A31 Row matrix Diagonal matrix AT = a b c d e f Matrix transpose A = a c e b d f

  3. a  c b  d Scalar multiplication and Matrix addition • If M = 1 2 3 • 4 5 6 • 3M = 3 6 9 • 12 15 18 How about this?? c a a  c d e f =  b d b

  4. Scalar products • We can use matrices to represent vectors and use matrix multiplication to generate their scalar and vector products • A = [a1, a2, a3], B = [b1, b2, b3] • A.B = a1 a2 a3 = a1b1 + a2b2 + a3b3 b1 b2 b3 1 2 3 1 . = 1 2

  5. Determinants of a Matrix • If A = a11 a12 • a21 a22 • |A| = a11.a12 – a21.a12 • Example • If A = 2 4 • -1 2 • |A| = ?

  6. Determinants of a Matrix • A = a11 a12 a13 • a21 a22 a23 • a31 a32 a33 • |A| = a11 a22 a23 – a12 a21 a23 + a13 a21 a22 • a32 a33 a31 a33 a31 a32 • Example • A = 4 0 -1 • 1 2 1 • -3 6 5

  7. Properties of determinants • |A| = |AT| • Interchanging any two rows or any two columns of A changes the sign of |A| • If we obtain B by multiplying one row or column of A by a constant, k then |B| = k|A| • If two rows or columns of A is identical, then |A| = 0 • If A square matrix and |A| = +1, it is orthogonal and proper. I |A| = -1, it is orthogonal and improper.

  8. Matrix inversion • The inverse of a square matrix A is A-1 • AA-1 = A -1A = I • If an inverse exists, the matrix is said to be a nonsingular matrix, otherwise the matrix is called a singular matrix. • Element of A-1are aij-1 where • aij-1 = (-1) i+j|Aji| • |A|

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