Suppose we have a 2 x 2 (square) matrix with elements \(a\), \(b\), \(c\), and \(d\) that are arranged in rows and columns.
$$A = \left[ {\matrix{a & b \cr c & d \cr } } \right]$$
The determinant of matrix A is calculated using the following formula.
$$\det \left( A\right) = ad - cb$$
The determinant is just the difference of the products of the diagonals. The first diagonal is the downward diagonal which is top-left [\(a\)] times bottom-right [\(d\)]. The second diagonal is the upward diagonal which is bottom-left [\(c\)] times top-right [\(b\)].
Example 1: Find the determinant of the 2x2 matrix.
$$B = \left[ {\matrix{2 & 5 \cr 3 & 9 \cr } } \right]$$
Let's calculate the determinant of matrix B using the formula. For the downward diagonal, we multiply the top-left (2) by the bottom-right (9) which gives us \(\left( 2 \right)\left( 9 \right) = 18\). For the upward diagonal, we multiply the bottom-left (3) by the top-right (5), that is, \(\left( 3 \right)\left( 5 \right) = 15\). Finally, we subtract the product of the downward diagonal by the product of the upward diagonal.
$$\eqalign{
\det \left( B \right) &= \left( 2 \right)\left( 9 \right) - \left( 3 \right)\left( 5 \right) \cr
& = 18 - 15 \cr
& = 3 \cr} $$
Example 2: Find the determinant of the 2x2 matrix.
$$C = \left[ {\matrix{ { - 6} & 7 \cr { - 2} & 1 \cr } } \right]$$
Be careful here when subtracting the product of the second diagonal. We have a case of minus a negative number.
$$\eqalign{
\det \left( C \right) &= \left( { - 6} \right)\left( 1 \right) - \left( { - 2} \right)\left( 7 \right) \cr
& = - 6 - \left( { - 14} \right) \cr
& = - 6 + 14 \cr
& = 8 \cr} $$
Example 3: Calculate the determinant of the 2x2 matrix.
$$D = \left[ {\matrix{ 3 & { - 2} \cr { - 8} & 4 \cr } } \right]$$
We know that \(a=3\), \(b=-2\), \(c=-8\), and \(d=4\). We substitute those values into the formula then simplify.
$$\det \left[ {\matrix{ a & b \cr c & d \cr } } \right] = ad - cb$$
$$\eqalign{
d\left( D \right) &= \left( 3 \right)\left( 4 \right) - \left( { - 8} \right)\left( { - 2} \right) \cr
& = 12 - 16 \cr
& = - 4 \cr} $$