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Q : 18       If A is an invertible matrix of order 2, then det   is equal to

(A)        (B)         (C)         (D)

Given that the matrix is invertible hence  exists and  Let us assume a matrix of the order of 2; . Then .     and   Now,     Taking determinant both sides; Therefore we get; Hence the correct answer is B.

Q : 17        Let A be a nonsingular square matrix of order . Then  is equal to

(A)       (B)       (C)       (D)

We know the identity  Hence we can determine the value of . Taking both sides determinant value we get,      or    or taking R.H.S., or, we have then      Therefore  Hence the correct answer is B.

Q : 16      If    , verify that . Hence find .

Given matrix: ; To show:  Finding each term: So now we have,  Now finding the inverse of A; Post-multiplying by  as,                                          ...................(1) Now, From equation (1) we get; Hence inverse of A is :

Q : 15     For the matrix     Show that      Hence, find .
.

Given matrix: ; To show:  Finding each term: So now we have,  Now finding the inverse of A; Post-multiplying by  as,                                          ...................(1) Now, From equation (1) we get;

Q : 14      For the matrix  , find the numbers  and  such that .

Given  then we have the relation   So, calculating each term; therefore  ; So, we have equations;     and  We get .

Q : 13            If   ​ , show that  . Hence find

Given  then we have to show the relation  So, calculating each term; therefore  ; Hence .   [Post multiplying by , also ]

Q : 12        Let      and  .  Verify that  .

We have  and . then calculating; Finding the inverse of AB. Calculating the cofactors fo AB:                Then we have adj(AB): and |AB| = 61(67) - (-87)(-47) = 4087-4089 = -2 Therefore we have inverse:                                                        .....................................(1) Now, calculating inverses of A and B. |A| = 15-14 = 1   and |B| = 54- 56 = -2      and ...

Q : 11       Find the inverse of each of the matrices (if it exists).

Given the matrix :   To find the inverse we have to first find adjA then as we know the relation: So, calculating |A| : Now, calculating the cofactors terms and then adjA.                                                                                       So, we have  Therefore inverse of A will be:

Q : 10         Find the inverse of each of the matrices (if it exists).

Given the matrix :   To find the inverse we have to first find adjA then as we know the relation: So, calculating |A| : Now, calculating the cofactors terms and then adjA.                                                                                       So, we have  Therefore inverse of A will be:

Q : 9         Find the inverse of each of the matrices (if it exists).

Given the matrix :   To find the inverse we have to first find adjA then as we know the relation: So, calculating |A| : Now, calculating the cofactors terms and then adjA.                                                                                       So, we have  Therefore inverse of A will be:

Q : 8       Find the inverse of each of the matrices (if it exists).

Given the matrix :   To find the inverse we have to first find adjA then as we know the relation: So, calculating |A| : Now, calculating the cofactors terms and then adjA.                                                                                       So, we have  Therefore inverse of A will be:

Q : 7       Find the inverse of each of the matrices (if it exists).

Given the matrix :   To find the inverse we have to first find adjA then as we know the relation: So, calculating |A| : Now, calculating the cofactors terms and then adjA.                                                                                       So, we have  Therefore inverse of A will be:

Q : 6         Find the inverse of each of the matrices (if it exists).

Given the matrix :   To find the inverse we have to first find adjA then as we know the relation: So, calculating |A| : |A| = (-2+15) = 13 Now, calculating the cofactors terms and then adjA. So, we have  Therefore inverse of A will be:

Q : 5          Find the inverse of each of the matrices (if it exists).

Given matrix :   To find the inverse we have to first find adjA then as we know the relation: So, calculating |A| : |A| = (6+8) = 14 Now, calculating the cofactors terms and then adjA. So, we have  Therefore inverse of A will be:

Q : 4          Verify .

Given matrix:  Let   Calculating the cofactors;            Hence,  Now,   also,   Now, calculating |A|;    So,  Hence we get,  .

Q : 3           Verify .

Given the matrix:  Let   Calculating the cofactors;     Hence,  Now,   aslo,   Now, calculating |A|;    So,  Hence we get

Q : 2     Find adjoint of each of the matrices

Given the matrix:  Then we have, Hence we get:

Q : 1       Find adjoint of each of the matrices.

Given matrix:  Then we have, Hence we get:
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