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Physics LibreTexts

66.17: Matrix Properties

( \newcommand{\kernel}{\mathrm{null}\,}\)

This appendix presents a brief summary of the properties of 2×2 and 3×3 matrices.

2×2 Matrices

Determinant

The determinant of a 2×2 matrix is given by the well-known formula:

det(abcd)=adbc

Matrix of Cofactors

The matrix of cofactors is the matrix of signed minors; for a 2×2 matrix, this is

cof(abcd)=(dcba)

Inverse

Finally, the inverse of a matrix is the transpose of the matrix of cofactors divided by the determinant. For a 2×2 matrix,

(abcd)1=1adbc(dbca)

3×3 Matrices

Determinant

The determinant of a 3×3 matrix is given by:

det(abcdefghi)=a(eifh)b(difg)+c(dheg)

Matrix of Cofactors

The matrix of cofactors is the matrix of signed minors; for a 3×3 matrix, this is

cof(abcdefghi)=(eifhfgdidhegchbiaicgbgahbfcecdafaebd)

Inverse

Finally, the inverse of a matrix is the transpose of the matrix of cofactors divided by the determinant. For a 3×3 matrix,

(abcdefghi)1=1a(eifh)b(difg)+c(dheg)(eifhchbibfcefgdiaicgcdafdhegbgahaebd)


66.17: Matrix Properties is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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