# Characteristic equation of a matrix and inverse of a matrix

Linear Algebra

Matrices

Eigen Values and Eigen Vectors(I)

Find the characteristic equation of the matrix

A = [1, 3, 7; 4, 2, 3; 1, 2, 1]

Show that the equation is satisfied by and hence obtain the inverse of the given matrix.

The fully formatted problem is in the attached file.

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Linear Algebra

Matrices

Eigen Values and Eigen Vectors(I)

By:- Thokchom Sarojkumar Sinha

Find the characteristic equation of the matrix

Show that the equation is satisfied by and hence obtain the inverse of the given matrix.

Solution:- The characteristic equation of is

i.e.,

or,

or,

or,

or,

or,

or.,

or, ----------------------------------(1)

which is the required characteristic equation of .

Since by Cayley Hamilton Theorem, every square matrix satisfies its own characteristic

equation.

Hence the matrix must satisfy its characteristic equation.

i.e., ----------------------------------(2)

where is the unit matrix of order 3.

We have to show the above relation.

L.H.S. of (2)

= R.H.S. of (2)

Hence shown.

Again, we have

Multiplying both sides of the above equation by , we get

or,

or,

Therefore

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