Determinant of a Matrix (2x2)

Determinant of a Matrix (2x2)

How to find the determinant of a matrix (2x2): formula, examples, and their solutions.

Formula

The determinant of a 2x2 matrix is ad - bc. It determines the properties of a matrix.

|a b / c d| means
the determinant of the matrix [a b / c d].

It determines
whether [a b / c d] has an inverse matrix.

Inverse matrix

For a 2×2 matrix [a b / c d],
|a b / c d| is equal to,

multiply the blue arrow elements, ad,
minus,
multiply the brown arrow elements, bc.

Example 1

Find the determinant of [1 2 / 3 4].

The determinant, |1 2 / 3 4|, is equal to,

multiply the blue arrow elements, 1⋅4
minus,
multiply the brown arrow elements, 2⋅3.

1⋅4 = 4
2⋅3 = 6

4 - 6 = -2

So the determinant is -2.

Example 2

Find the value of x. det[2 7 / 3 x] = 5.

det[2 7 / 3 x] means the determinant: |2 7 / 3 x|.

So the determinant, |2 7 / 3 x|, is equal to,

multiply the blue arrow elements, 2⋅x
minus,
multiply the brown arrow elements, 7⋅3.

7⋅3 = 21

|2 7 / 3 x| = 2x - 21

And it says
det[2 7 / 3 x] = 5.

So the right side is 5.

Write 2x - 21 = 5
and solve the equation.

Move -21 to the right side.

Then 2x = 26.

Divide both sides by 2.

Then x = 13.