Adjoint and Inverse of a Matrix

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Adjoint of a Square Matrix

The adjoint of a square matrix [a_ij] is defined as the transpose of the matrix [A_ij] where A_ij are the cofactors of the elements a_ij.

Adjoint of A is denoted by adj A.

Example:

Find the adjoint of the matrix.

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 A.

Note 1:

Only a square matrix can have its inverse.

Note 2:

From the definition, it is clear that if B is the inverse of A, then A is the inverse of B.

Note 3:

Inverse of A is denoted by A^-1, thus B = A^-1 and

AA^-1 = A^-1A=I.

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 = C

Hence the inverse of A is unique.

Theorem

If A and B are two invertible matrices of the same order, then (AB)^-1 = B^-1A^-1.

Proof:

From the definition of inverse of a matrix, we have

(AB)(AB)^-1 = I

or A^-1 (AB)(AB)^-1 = A^-1 I (Pre-multiplying both sides by A^-1)

or (A^-1A) B (AB)^-1 = A^-1 (Since A^-1 I = A^-1)

or I B (AB)^-1 = A^-1

or B (AB)^-1 = A^-1

or (B^-1B)(AB)^-1 =B^-1A^-1

or I(AB)^-1= B^-1A^-1

or (AB)^-1 = B^-1A^-1

Theorem

Elementary Transformation

Elementary transformations are of the following three types:

• Interchange of any two rows (or columns)

• The multiplication of the elements of a row (or column) by a non-zero number.

• The addition to the elements of any row (or column) the corresponding elements of any other row (or column) multiplied by any number.

Any elementary operation is called a row transformation or a column transformation according as it applies to rows or columns.

Definition

Let R_i denotes the i^th row of the matrix A = [a_ij] then the elementary row operations on the matrix A are defined as:

3. R_i g R_i + kR_j means multiply each element of j^th row by k and add it to the corresponding elements of i^th row.

The corresponding column transformations are

Properties of adjoint of a matrix

1. A.(adj A) = (adj A). A = |A| I

2. adj (AB) = (adj B) . (adj A).

Non-singular Matrix

A square matrix A is said to be non-singular if its determinant value is non-zero.

i.e.,

Singular Matrix

A square matrix A is said to be singular if |A| = 0.

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 also non-singular matrices of the same order.

Consistency and Inconsistency of a System of Linear Equations

A system of linear equations is said to be consistent if it has a solution. This means that the solution satisfies all the equations in the system simultaneously.

If a system of linear equations has no solution, then it is said to be inconsistent.