Unit – 2A | unit 2a matrices

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Unit – 2A

Matrices

Q1) Find the inverse of matrix A given below:

A1) Let A = IA

Thus, the inverse of matrix A is given by I =

Therefore,

Q2) Find the inverse of matrix A given below:

A2) Let A = IA

Thus, the inverse of matrix A is given by I =

Therefore,

Q3) Reduce the following matrix into normal form and find its rank,

A3) Let A =

Apply we get

Apply

Hence the rank of matrix A is 2 i.e. .

Q4) Reduce the following matrix into normal form and find its rank,

A4) Let A =

Apply and

Apply

Hence the rank of the matrix A is 2 i.e. .

Q5) Reduce the following matrix into normal form and find its rank,

A5) Let A =

Apply

Apply and

Apply

Hence the rank of matrix A is 2 i.e. .

Q6) Solve the equations

4x+7y-9=0

5x-8y+15=0

A6) Given equation can be written in matrix form as A = , X= ,

B =

Given system can be written as: AX=B, where x= A-1 B

Let us find determinant :|A| = 4*(-8)-(5*7) = -32-35 = -67 So, solution exist.

Minor and co-factor of matrix A are:

Cofactor matrix =

X =

X = and Y =

Q7) Solve the system of equations:

2x+y+3z = 1, x+z =2, 2x+y+z= 3

A7) Given equation can be written in matrix form as A =

X = , B =

Given system of equations can be written as AX = B where X = A-1B

Let us find determinant :|A| = 2(0-1)-1(1-2) +3(1-0) = -2+1+3 = 2.

So, solution exist.

Cofactor = and Adj(A) =

X = 3, Y = -2, Z = -1.

Q8) Let us test whether the given matrices are symmetric or not i.e., we check for,

A =

A8) Now

A =

Hence the given matric symmetric

Example 8: let A be a real symmetric matrix whose diagonal entries are all positive real numbers.

Is this true for all of the diagonal entries of the inverse matrix A-1 are also positive? If so, prove it. Otherwise give a counter example

A) The statement is false, hence we give a counter example

Let us consider the following 22 matrix

A =

The matrix A satisfies the required conditions, that is A is symmetric and its diagonal entries are positive.

The determinant det(A) = (1)(1)-(2)(2) = -3 and the inverse of A is given by

A-1= =

By the formula for the inverse matrix for 22 matrices.

This shows that the diagonal entries of the inverse matric A-1 are negative.

Skew-symmetric: A skew-symmetric matrix is a square matrix that is equal to the negative of its own transpose. Anti-symmetric matrices are commonly called as skew-symmetric matrices.

Q9) Let A and B be nn skew-matrices. Namely AT = -A and BT = -B

(a) Prove that A+B is skew-symmetric.

(b) Prove that cA is skew-symmetric for any scalar c.

A9) (a) (A+B)T = AT + BT = (-A) +(-B) = -(A+B)

Hence A+B is skew symmetric.

(b) (cA)T = c.AT =c(-A) = -cA

Thus, cA is skew-symmetric.

(c)Let P be an mn matrix. Prove that PT AP is skew-symmetric.

Using the properties, we get,

(PT AP)T = PTAT(PT)T = PTATp

= PT (-A) P = - (PT AP)

Thus (PT AP) is skew-symmetric.

Orthogonal matrix: An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. Although we consider only real matrices here, the definition can be used for matrices with entries from any field.

Suppose A is a square matrix with real elements and of n x n order and AT is the transpose of A. Then according to the definition, if, AT = A-1 is satisfied, then,

A AT = I

Where ‘I’ is the identity matrix, A-1 is the inverse of matrix A, and ‘n’ denotes the number of rows and columns.

Q10) Prove Q= is an orthogonal matrix

A10) Given Q=

So, QT = …..(1)

Now, we have to prove QT = Q-1

Now we find Q-1

Q-1 =

Q-1 = … (2)

Now, compare (1) and (2) we get QT = Q-1

Therefore, Qis an orthogonal matrix.

Q11) Let us consider the following 3 ×3 matrix

A =

A11) We want to diagonalize the matrix if possible.

Step 1: Find the characteristic polynomial

The Characteristic polynomial p(t) of A is

Using the cofactor expansion, we get

Step2: From the characteristic polynomial obtained step1, we see that eigenvalues are

And

Step2: Find the eigenspaces

Let us first find the eigenspaces corresponding to the eigenvalue

By definition, is the null space of the matrix

By elementary row operations.

Hence if (A-I) x=0 for x

Therefore, we have

From this, we see that the set

Is a basis for the eigenvalues

Thus, the dimension of , which is the geometric multiplicity of

Similarly, we find a basis of the eigenspaces for the eigenvalue We have

By elementary row operations.

Then if (A-2I) x=0 for x

Therefore, we obtain

From this we see that the set

and the geometric multiplicity is 1.

Since for both eigenvalues, the geometric multiplicity is equal to the algebraic multiplicity, the matrix A is not defective, and hence diagonalizable.

Step4: Determine linearly independent eigenvectors

From step3, the vectors

Are linearly independent eigenvectors.

Step5: Define the invertible matrix S

Define the matrix S=

And the matrix S is non-singular (since the column vectors are linearly independent).

Step 6: Define the diagonal matrix D

Define the diagonal matrix

Note that (1,1) entry of D is 1 because the first column vector

Of S is in the eigenvalues that is is an eigenvector corresponding to eigenvalue

Similarly, the (2,2) entry of D is 1 because the second column of S is in .

The (3,3) entry of D is 2 because the third column vector of S is in

(The order you arrange the vector to form S does not matter but once you made S, then the order of the diagonal entries is determined by S, that is , the order of eigenvectors on S)