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66.17: Matrix Properties

Last updated

Feb 4, 2024
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- 66.16: Vector Arithmetic
- 66.18: Newton’s Laws of Motion (Original)

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This appendix presents a brief summary of the properties of $2 \times 2$ and $3 \times 3$ matrices.

$2 \times 2$ Matrices

Determinant

The determinant of a $2 \times 2$ matrix is given by the well-known formula:

$\operatorname{det}\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)=a d-b c \text {. }$

Matrix of Cofactors

The matrix of cofactors is the matrix of signed minors; for a $2 \times 2$ matrix, this is

$\operatorname{cof}\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)=\left(\begin{array}{cc} d & -c \\ -b & a \end{array}\right)$

Inverse

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

$\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)^{-1}=\frac{1}{a d-b c}\left(\begin{array}{cc} d & -b \\ -c & a \end{array}\right)$

$3 \times 3$ Matrices

Determinant

The determinant of a $3 \times 3$ matrix is given by:

$\operatorname{det}\left(\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right)=a(e i-f h)-b(d i-f g)+c(d h-e g)$

Matrix of Cofactors

The matrix of cofactors is the matrix of signed minors; for a $3 \times 3$ matrix, this is

$\operatorname{cof}\left(\begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array}\right)=\left(\begin{array}{ccc} e i-f h & f g-d i & d h-e g \\ c h-b i & a i-c g & b g-a h \\ b f-c e & c d-a f & a e-b d \end{array}\right)$

Inverse

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

$\left(\begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array}\right)^{-1}=\frac{1}{a(e i-f h)-b(d i-f g)+c(d h-e g)}\left(\begin{array}{ccc} e i-f h & c h-b i & b f-c e \\ f g-d i & a i-c g & c d-a f \\ d h-e g & b g-a h & a e-b d \end{array}\right)$

2×22 \times 2 Matrices

Determinant

Matrix of Cofactors

Inverse

3×33 \times 3 Matrices

Determinant

Matrix of Cofactors

Inverse

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$2 \times 2$ Matrices

$3 \times 3$ Matrices