2x2 matrix determinant and inverse calculator
Everything about a 2x2 matrix follows from its determinant and trace.
Determinant
10
- Trace
- 10
- Inverse, row 1 column 1
- 0.6
- Inverse, row 1 column 2
- -0.7
- Inverse, row 2 column 1
- -0.2
- Inverse, row 2 column 2
- 0.4
- First eigenvalue
- 8.872 983 346 21
- Second eigenvalue
- 1.127 016 653 79
Show the working
- a (row 1, column 1) = 4
- b (row 1, column 2) = 7
- c (row 2, column 1) = 2
- d (row 2, column 2) = 6
- Determinant = 4*6 - 7*2 = 10
- Trace = 4 + 6 = 10
- Inverse, row 1 column 1 = 10 != 0 ? 6/10 : 0 = 0.6
- Inverse, row 1 column 2 = 10 != 0 ? -7/10 : 0 = -0.7
- Inverse, row 2 column 1 = 10 != 0 ? -2/10 : 0 = -0.2
- Inverse, row 2 column 2 = 10 != 0 ? 4/10 : 0 = 0.4
- disc = 10^2 - 4*10 = 60
- First eigenvalue = 60 >= 0 ? (10 + sqrt(60))/2 : 0 = 8.872 98
- Second eigenvalue = 60 >= 0 ? (10 - sqrt(60))/2 : 0 = 1.127 02
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- det = ad - bc
- inverse = (1/det) x [[d, -b], [-c, a]]
- eigenvalues = (trace +/- sqrt(trace^2 - 4 det)) / 2
A zero determinant means the matrix is singular and has no inverse; the eigenvalue fields read zero when the eigenvalues are complex.