The element in the first row and third column of the below inverse matrix is:

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answer is D.

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Detailed Solution

Let us assume that the given matrix as

A = 1 2 − 3 0 1 2 0 0 1

Now, let us find the minor matrix of the given matrix
Let us assume that the minor matrix of the given matrix as M
We know that minors of each element are obtained by the determinant of matrix eliminating the column and row of the element
By using the above condition to the element of first row first column then we get

⇒ m 1 = 1201 ⇒ m 1 = 1 × 1 − 0 × 2 ⇒ m 1 = 1

Similarly by using the above condition to all elements of the given matrix we get
Now, lest us find the cofactor matrix of the above miner matrix
Let us assume that the cofactor matrix as C
We know that the cofactor matrix is obtained by giving the alternative negative to mince matrix then we get

⇒ C = 1 − 0 0 − 2 1 − 0 7 − 2 1 ⇒ C = 100 − 2 10 7 − 2 1

Now, les us find the adjoint matrix
We know that the adjoint matrix is given as the transpose of the cofactor matrix
By using the above condition we get

⇒ a d j A = 1 − 0 0 − 2 10 7 − 2 1 T

⇒ a d j A = 1 − 2 7 01 − 2 001

Now, let us find the determinant of the given matrix

⇒ | A | = 12 − 3 012001

By expanding the determinant along first column then we get

⇒ | A | = 1 (1 × 1 − 0 × 2) − 0 (2 × 1 − 0 × (− 3)) + 0 (2 × 2 − 1 × (− 3)) ⇒ | A | = 1

We know that the condition that inverse of matrix is given as

A − 1 = a d A | A |

By using the above formula we get the inverse of the given matrix as

⇒ A − 1 = 11 1 − 2 7 01 − 2 001 ⇒ A − 1 = 1 − 2 7 01 − 2 001

We are asked to find the element in the first row and third column of the inverse matrix of a given matrix.
Here, we can see that the element in the first row and the second column is 7

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