 A matrix is defined as a rectangular array of elements. |
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| If the arrangement has m rows and n columns, then the matrix is of order mxn (read as m by n). |
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| A matrix is enclosed by a pair of parameters such as ( ) or [ ]. It is denoted by a capital letter. |
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| Two matrices are said to be comparable if they have the same order. |
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| Addition and subtraction of two matrices is possible only if they have the same order. |
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| If two matrices A and B are of same order, then A - B = A + (- B). |
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| Commutative law, associative law holds good for addition of matrices. |
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| The additive identity of a matrix A of order mxn is the zero matrix of order mxn. |
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| The additive inverse of a matrix A is -A. |
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| The multiplication of two matrices A and B is possible if the number of columns of A is equal to the number of rows B. |
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| Suppose A is a matrix of order mxn and B is a matrix of order nxp, the matrix AB is of order mxp. |
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| If A, B and C are the matrices which can be multiplied then |
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| (a) Matrix multiplication is not commutative, |
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i.e., AB  BA (always) |
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| (b) Associative law holds good for matrix multiplication, |
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| i.e., (AB)C = A(BC) |
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| (c) Matrix multiplication is distributive with respect to addition |
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| A(B + C) = AB + AC |
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| or (A + B)C = AC + BC |
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| If A is a matrix of order mxn and is a scalar (real or complex) then the matrix kA is obtained by multiplying each element of A by k. |
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| To every square matrix, a value can be associated which is known as the determinant of the matrix. |
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| Note that the determinant of kA where k is a scalar and A is a square matrix, is given by kn times determinant of A. |
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| i.e., is |kA| = kn |A| |
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| The value of the determinant remain unchanged if its rows and columns are interchanged |
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| If two rows or columns of a determinant are interchanged, then the sign of the determinant is changed. |
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| If any two rows or columns of a determinant are equal, then its value is zero. |
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| If each element of a row or column of a determinant multiplied by k, then its value is multiplied by k. |
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| If two rows or columns of determinant are proportional, the value of the determinant is zero. |
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| A square A = [aij] is said to be symmetric if AT = A, i.e., if |
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| aij = aji |
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| A square matrix A is said skew symmetric if AT = - A, |
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| i.e., aij = - aji |
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| Any square matrix A can be expressed as the sum of a symmetric matrix and a skew symmetric matrix as follows |
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| For a 2 x 2 matrix, the adjoint is got by interchanging elements in the leading diagonal and changing signs in the other diagonal. |
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| If A =[aij]mxn is a matrix of order mxn. The minor of aij of |A|, denoted by Mij, is given by the determinant which is obtained by deleting ith row jth column of |A|. |
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| The co-factor of the determinant of the A = [aij]mxn, denoted by Aij is given by |
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| Aij = (-1)i+j Mij |
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| The transpose of a matrix A, denoted by AT, is obtained by interchanging the rows and columns of A. |
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| The adjoint of a square matrix A = [aij] is defined as the transpose of the matrix [Aij] where Aij is the co-factor of the element aij. |
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| Adjoint of A is denoted by Adj A. |
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| Note that the concept of adj is only for square matrix. |
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A square matrix A is said to be non-singular if |A|  0. |
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| Let A be a square matrix of order n. If there exists a square matrix B of order n, such that AB = BA = In, where In is the identify matrix of order n, then B is called the inverse of A. |
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The inverse of a matrix A exists if and only if |A|  0. |
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| In other words, every non-singular matrix is invertible. |
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| The area of a triangle whose vertices are (x1, y1), (x2, y2) and |
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 |
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| The following are the steps to solve a system of linear equations |
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| using Cramer's rule. |
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| Step 1: Find the value of the determinant |
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Step 2: If D  0, then the system has unique solution, given by |
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 |
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| Where D1, D2 and D3 are the determinants obtained from D by replacing respectively the first column, 2nd column and third column containing the constant terms d1, d2, d3. |
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| Step 3: If D = 0, the system may have infinite number of solutions or no solution. |
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| A system of linear equations is said to be consistent if it has at least are solution, otherwise it is inconsistent. |
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| Let A be asquare matrix of order n. Following are the steps to find the inverse of a matrix. |
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| Step 1: Find the value of the determinants A. That is, find |A| |
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| Step 2: If |A| = 0, inverse of the matrix A does not exists. |
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Step 3: If |A|  0, find the co-factors Aij of all the elements of A. |
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| Step 4: Find adj A, the transpose of the matrix of co-factors Aij. |
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| Step 5: |
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| Following are the steps to solve a system of linear equations with three unknown, using inverse of a matrix (Matrix method) |
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| Let the given system of equations be |
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| Step 1: |
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| The system of linear equations may be expressed as AX = B. |
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Step 2: Find |A|. If |A|  0, the system has unique solution which is given by X = A-1B. |
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| Step 3: |
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| If |A| = 0, put x = k (y = k or z = k) in any two of the given equations and find y and z in terms of k. |
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| Substitute these values of x, y and z in terms of k in the third equation. If the third equation is satisfied by these values of x, y and z, then the system has infinitely many solutions. |
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| If the third equation is not satisfied, the system has no solution. |