Square Matrix
A matrix in which the number of rows is equal to the number of columns, say n, is called a square matrix of order n. In this square matrix of order n the elements a 1 1 , a 2 2 .......a n n is called the principal diagonal or the leading diagonal. The elements a 1 1 , a 2 2 ,.....
A matrix in which the number of rows is equal to the number of columns, say n, is called a square matrix of order n. In this square matrix of order n the elements a 1 1 , a 2 2 .......a n n is called the principal diagonal or the leading diagonal. The elements a 1 1 , a 2 2 ,.....Inverse of a Square Matrix
Let A be a square matrix of order n. If there exists a matrix B of order n such that AB = BA = I, where I is the identity matrix of order n, then the matrix A is said to be invertible and B is called the inverse (or reciprocal) of ..
Matrices and Determinants
Matrices and Determinants..
Matrices and Determinants..Examples:
skew-symmetric for every square matrix A. That is any square matrix is expressible as the sum of a symmetric matrix and a skew-symmetric matrix. 5. A matrix which is both symmetric and skew symmetric is a zero matr..
skew-symmetric for every square matrix A. That is any square matrix is expressible as the sum of a symmetric matrix and a skew-symmetric matrix. 5. A matrix which is both symmetric and skew symmetric is a zero matr..Note 1:
Only a square matrix can have its invers..
Determinants
Let A = [aij] be a square matrix. We can associate with the square matrix A, a determinant which is formed by exactly the same array of elements of the matrix A. A determinant formed by the same array of elements of the square matrix A is called the determinant ..
Symmetric Matrix
A square matrix A = [a i j ] is said to be symmetric if its (i,j) t h element is the same as its (j,i) t h element. i.e a i j = a j i " i, j A square matrix A is said to be symmetric, if A = A ..
A square matrix A = [a i j ] is said to be symmetric if its (i,j) t h element is the same as its (j,i) t h element. i.e a i j = a j i " i, j A square matrix A is said to be symmetric, if A = A ..Determinants
Let A = [a ij ] be a square matrix. We can associate with the square matrix A, a determinant which is formed by exactly the same array of elements of the matrix A. A determinant formed by the same array of elements of the square matrix A is called the determinant of th..
Let A = [a ij ] be a square matrix. We can associate with the square matrix A, a determinant which is formed by exactly the same array of elements of the matrix A. A determinant formed by the same array of elements of the square matrix A is called the determinant of th..Theorem:
The inverse of a square matrix if it exists, is unique. Let A be an invertible square matrix. If possible, let B and C be two inverse of A. Then AB = BA = I. AC = CA = I (by def. of inverse) Now, B = BI = B(AC) = (BA)C [ Matrix multiplication is associative] = IC = C i.e., B =..
The inverse of a square matrix if it exists, is unique. Let A be an invertible square matrix. If possible, let B and C be two inverse of A. Then AB = BA = I. AC = CA = I (by def. of inverse) Now, B = BI = B(AC) = (BA)C [ Matrix multiplication is associative] = IC = C i.e., B =..Adjoint and Inverse of a Matrix
Adjoint of a Square Matrix - The adjoint of a square matrix [a i j ] is defined as the transpose of the matrix [A i j ] where A i j are the cofactors of the elements a i j . Adjoint of A is denoted by adj ..
Adjoint of a Square Matrix - The adjoint of a square matrix [a i j ] is defined as the transpose of the matrix [A i j ] where A i j are the cofactors of the elements a i j . Adjoint of A is denoted by adj ..See what our Users say :
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