Question

In: Math

Determine the Kernel of matrix B. Determine the Cokernel of matrix B.

Matrix A
1	2	1	3
1	-1	1	2
2	4	1	3
3	2	1	-1

Matrix B
1	2	3	-2	-1	2
2	-1	2	3	2	-3
3	1	2	-1	3	-5
5	5	5	-7	7	-4

Determine the Kernel of matrix B.
Determine the Cokernel of matrix B.
Determine the row space of the Cokernel of B.
Determine the inverse of matrix A.

Perform LU (or PLU) factorization on matrix A and determine the L and U (and/or P) and E matrices.
Find the value of the four variables in matrix A using Cramer's rule.
Choose a row or column and use the elements to represent matrix A as a series of cofactors.

Solutions

Expert Solution

Solution : ( 1 )

Kernel of matrix ( B ) :

**Matrix B**
1	2	3	-2	-1	2
2	-1	2	3	2	-3
3	1	2	-1	3	-5
5	5	5	-7	7	-4

We know that kernel of matrix = nullspace of matrix

So, we can find out the nullspace of matrix.

We begin with the matrix:

Add -2 times row 1 to row 2:

Add -3 times row 1 to row 3:

Add -5 times row 1 to row 4:

Multiply row 2 by -1/5:

Add 5 times row 2 to row 3:

Add 5 times row 2 to row 4:

Multiply row 3 by -1/3:

Add 6 times row 3 to row 4:

Multiply row 4 by 1/4:

Add 2/3 times row 4 to row 3:

Add 4/5 times row 4 to row 2:

Add 1 times row 4 to row 1:

Add -4/5 times row 3 to row 2:

Add -3 times row 3 to row 1:

Add -2 times row 2 to row 1:

Convert the matrix equation back to an equivalent system:

Add an equation for each free variable:

Solve for each variable in terms of the free variables:

Collect terms into vectors:

Factor out variables on the right side:

Thus, a basis for the null space is:

Hence,

Kernel of matrix ( B ) :

-----------------------------------------------------------------------------------------------------------------------------------------------------------

Solution : ( 4 )

Inverse of matrix A :

To find the inverse, we form an augmented matrix by appending an identity matrix to the right, and then put this matrix in reduced row-echelon form.

We begin with the matrix:

Add -1 times row 1 to row 2:

Add -2 times row 1 to row 3:

Add -3 times row 1 to row 4:

Multiply row 2 by -1/3:

Add 4 times row 2 to row 4:

Multiply row 3 by -1:

Add 2 times row 3 to row 4:

Multiply row 4 by -3/8:

Add -3 times row 4 to row 3:

Add -1/3 times row 4 to row 2:

Add -3 times row 4 to row 1:

Add -1 times row 3 to row 1:

Add -2 times row 2 to row 1:

The inverse is the right half of this augmented matrix:

milcah answered 2 years ago

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