2 MatricesA matrix is a rectangular arrangement of numbers in rows and columns. Rows run horizontally and columns run vertically.The dimensions of a matrix are stated “m x n” where ‘m’ is the number of rows and ‘n’ is the number of columns.
3 Equal MatricesTwo matrices are considered equal if they have the same number of rows and columns (the same dimensions) AND all their corresponding elements are exactly the same.
5 Matrix AdditionYou can add or subtract matrices if they have the same dimensions (same number of rows and columns).To do this, you add (or subtract) the corresponding numbers (numbers in the same positions).
10 Matrix Multiplication Matrix Multiplication is NOT Commutative! Order matters!You can multiply matrices only if the number of columns in the first matrix equals the number of rows in the second matrix.2 columns2 rows
11 Matrix Multiplication Take the numbers in the first row of matrix #1. Multiply each number by its corresponding number in the first column of matrix #2. Total these products.The result, 11, goes in row 1, column 1 of the answer. Repeat with row 1, column 2; row 1 column 3; row 2, column 1; ...
12 Matrix Multiplication Notice the dimensions of the matrices and their product.3 x 22 x 33 x 3________
13 Matrix Multiplication Another example:3 x 22 x 13 x 1
14 Properties of Matrix Multiplication Matrix Multiplication is not commutative, i.e. AB ≠ BAMatrix Multiplication is associative, i.e. A(BC) = (AB)CMatrix Multiplication is distributive, i.e. A(B+C) = AB+AC
15 Special Types of Matrices Idempotent MatrixNilpotent MatrixInvolutory Matrix
16 Transpose of MatrixLet A be any matrix. The matrix obtained by interchanging rows and columns of A is called the transpose of A and is denoted by A’ or AT.
17 Properties of Transpose of Matrices The transpose of transposed matrix is equal to the matrix itself, i.e. (A’)’ = A.The transpose of the sum of the two matrices is equal to the transpose of the matrices, i.e. (A+B)’ = A’+B’.The transpose of the product of two matrices is equal to the product of their transposes in the reverse order, i.e. (AB)’ = B’A’.
18 Matrix DeterminantsA Determinant is a real number associated with a matrix. Only SQUARE matrices have a determinant.The symbol for a determinant can be the phrase “det” in front of a matrix variable, det(A); or vertical bars around a matrix, |A| or
20 Determinant of a 3x3 matrix Imagine crossing out the first row.And the first column.Now take the double-crossed element. . .And multiply it by the determinant of the remaining 2x2 matrix-382-¾418011
21 Determinant of a 3x3 matrix Now keep the first row crossed.Cross out the second column.Now take the negative of the double-crossed element.And multiply it by the determinant of the remaining 2x2 matrix.Add it to the previous result.-382-¾418011
22 Determinant of a 3x3 matrix Finally, cross out first row and last column.Now take the double-crossed element.Multiply it by the determinant of the remaining 2x2 matrix.Then add it to the previous piece.-382-¾418011
23 ComputationMethod of CofactorsAlso known as the expansion of minors
24 Method of MinorsDeterminant of a 2 x 2 matrix is difference in products of diagonal elements.
33 Determinant obtained by expanding along any row or column of matrix of cofactors Determinant of A given by
34 Determinant of A Element Minor Cofactor Element x Cofactor a11 = 1 7 -5-10a13 = 3-4-12Determinant of A = -15
35 Determinants of 4 x 4 matrices Computational energy increases as order of matrix increasesUse pivotal condensation (computer algorithm)
36 Key Properties of Determinant Determinant of matrix and its transpose are equal.If any two adjacent rows(columns) of a determinant are interchanged, the value of the determinant changes only in sign.If any two rows or two columns of a determinant are identical or are multiple of each other, then the value of the determinant is zero.If all the elements of any row or column of a determinant are zero, then the value of the determinant is zero.
37 If all the elements of any row (or column) of a determinant are multiplied by a quantity (K), the value of the determinant is multiplied by the same quantity.If each element of a row (or column) of a determinant is sum of two elements, the determinant can be expressed as the sum of two determinants of the same order.The addition of a constant multiple of one row (or column) to another row (or column) leave the determinant unchanged.The determinant of the product of two matrices of the same order is equal to the product of individual determinants.
38 Adjoint of a MatrixIf A is any square matrix, then the adjoint of A is defined as the transpose of the matrix obtained by replacing the element of A by their corresponding co-factors.Adj.A = Transpose of the cofactor matrix
39 Inverse MatrixInverse of square matrix A is a matrix A-1 that satisfies the following equationAA-1 = A-1A = I
40 Steps to success in Matrix Inversion If the determinant = 0, the inverse does not exist if the matrix is singular.Replace each element of matrix A, by it’s minorCreate the matrix of cofactorsTranspose the matrix of cofactorsForms the adjointDivide each element of the adjoint by the determinant of A.
41 Matrix InversionPre multiplying both sides of the last equation by A-1, and using the result that A-1A=I,we can getThis is one way to invert matrix A!!!
43 Properties of Inverse Matrices If A and B are non-singular matrices of the same order, then (AB)-1 = B-1.A-1The inverse of the transpose of a matrix is equal to the transpose of the inverse of that matrix, i.e. (A’)-1 = (A-1)’The inverse of the inverse of a matrix is the matrix itself i.e. (A-1)-1 = A
44 Cramer’s Rule Given an equation system Ax=d where A is n x n. |A1| is a new determinant were we replace the first column of |A| by the column vector d but keep all the other columns intact
45 Cramer’s RuleThe expansion of the |A1| by its first column (the d column) will yield the expressionbecause the elements di now take the place of elements aij.
46 Cramer’s RuleIn general,This is the statement of Cramers’Rule
48 Cramer’s Rule Find the solution of the equation system: ♫ Work this out!!!!
49 Cramer’s Rule Solutions: Note that |A| ≠ 0 is necessary condition for the application of Cramer’s Rule. Cramer’s rule is based upon the concept of the inverse matrix, even though in practice it bypasses the process of matrix inversion.
50 Rank of a Matrix The number ‘r’ is called the rank of the matrix A if There exists at atleast one non-zero minor of order r of AEvery minor of order (r+1) of A is zero.The rank of a matrix A is denoted by p(A).
51 Steps to Find Rank of a Matrix The given matrix should be a square matrix. If it is not so, the matrix should be made a square matrix by deleting the extra row or the column.Find the determinant of the square matrix given or obtained after deleting extra row or column.If determinant of the matrix is zero, then take the sub-matrix of the given matrix. Of the determinant of any one of the sub-matrices is not zero, then the order of that sub matrix would be the rank of the given matrix.