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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 ..
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 ..
Adjoint and Inverse of a Matrix
The adjoint of a square matrix [aij] is defined as the transpose of the matrix [Aij] where Aij are the cofactors of the elements aij. Adjoint of A is denoted by adj A. Let A be a square matrix of order n. If there exists a matrix B of order n such that AB = BA = I, ..
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..Properties of Inverse of Matrix
In other words, a square matrix A is invertible if and only if A is a non-singular matrix. (c) If A and B are invertible square matrices, then (AB) - 1 = B - 1 A - 1 (d) If A and B are two non-singular square matrices of the same order, then AB and BA are al..
In other words, a square matrix A is invertible if and only if A is a non-singular matrix. (c) If A and B are invertible square matrices, then (AB) - 1 = B - 1 A - 1 (d) If A and B are two non-singular square matrices of the same order, then AB and BA are al..Definition of a Matrix
A rectangular array of entries is called a Matrix. The entries may be real, complex or functions. The entries are also called as the elements of the matrix. The rectangular array of entries are enclosed in an ordinary bracket or in square bracket. Matrices are denoted by capital lette..
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 =..Non-singular Matrix
A square matrix A is said to be non-singular if its determinant value is non-zero. i.e.,..
A square matrix A is said to be non-singular if its determinant value is non-zero. i.e.,..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..See what our Users say :
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