Question

Answer all question plz !!!!!!!!! with formula !!!!!!!!!!!!!!!!!!!!!!!!!!!!!

4. Test whether the following matrices are nonsingular: 7 -1 0 (c)11 4 13 -3 -4 (a)19 3 -4 9 5 (d)3 0 1 10 8 6 4 -2 1 (b)-5 6 0 5. What can you conclude about the rank of each matrix in Prob. 4? 7. Rewrite the simple nationa-income model (3.23) in the Ax d format (with Y as the first variable in the vector x), and then test whether the coefficient matrix A is nonsingular.

Answer 1

4)

A matrix will be singular when it's determinant is zero.

A)

4	0	1
19	1	-3
7	1	0

Expanding along 1st row.

= 4*(1*0 - (-3)*1) - 0*(19*0 - 7*(-3)) + 1*(19*1 - 7*1)

= 12 + 0 + 12 = 24; Hence Non-singular

B)

4	-2	1
-5	6	0
7	0	3

Expanding along 1st row

= 4*(6*3 - 0*0) -(-2)*( -5*3 -7*0) +1*(-5*0 - 6*7)

= 72 - 30 -42

=0; Hence Singular.

C)

7	-1	0
1	1	4
13	-3	-4

Expanding along 1st row

= 7*(1*(-4) - 4*(-3)) - (-1)*(1*(-4) - 4*13) + 0*(1*(-3) - 1*13)

= 56 - 56 + 0

=0; Hence Singular

D)

-4	9	5
3	0	1
10	8	6

Expanding along 1st row

= (-4)(0*6 - 8*1) - 9*(6*3 -10*1) + 5*( 8*3 - 10*0)

= 32 -72 + 120

= 82 ; hence Non-singular

5)

A square matrix $A_{n*n}$ has rank n only if its determinant of order n is non-zero i.e. matrix of order n is non- singular.

From the above, we can see that A and D are a non-singular matrix and hence their rank is 3.

For B and C, we are sure that they has rank < 3 because they are the singular matrix of order 3.