We need to find the cofactor of a₁₂ in the matrix

A cofactor is the number you get when you remove the column and row of a designated element in a matrix, which is just a numerical grid in the form of a rectangle or a square. The cofactor is always preceded by a positive (+) or negative (-) sign, depending whether the element is in a + or - position. It is

Let us recall how to find the cofactor of any element:

If we are given with,

Cofactor of any element, say a₁₁ is found by eliminating first row and first column.

⇒ Cofactor of a₁₁ = a₂₂ × a₃₃ – a₂₃ × a₃₂

The sign of cofactor of a₁₁ is (+).

And, cofactor of any element, say a₁₂ is found by eliminating first row and second column.

⇒ Cofactor of a₁₂ = a₂₁ × a₃₃ – a₂₃ × a₃₁

The sign of cofactor of a₁₂ is (-).

Similarly,

First know what the element at position a₁₂ in the matrix is.

In ,

a₁₂ = -3

And as discussed above, the sign at a₁₂ is (-).

For cofactor of -3, eliminate first row and second column in the matrix.

⇒ Cofactor of -3 = (6 × -7) – (4 × 1)

⇒ Cofactor of -3 = -42 – 4

⇒ Cofactor of -3 = -46

Since, the sign of cofactor of -3 is (-), then

Cofactor of -3 = -(-46)

⇒ Cofactor of -3 = 46

Thus, cofactor of -3 is 46.