Is the matrix invertible? Matrix A nxn is invertible if there exists - - PowerPoint PPT Presentation

▶

Oct 20, 2023 224 likes •308 views

Matrix Inverse size 3x3 M. A. Hameed ECSE Department Rensselaer Polytechnic Institute Intro to ECSE Is the matrix invertible? Matrix A nxn is invertible if there exists B nxn such that AB = BA = I n Invertible = nonsingular =

SLIDE 1

Matrix Inverse size 3x3

M. A. Hameed

ECSE Department Rensselaer Polytechnic Institute

Intro to ECSE

SLIDE 2

M. Hameed
Matrix Anxn is invertible if there exists Bnxn such

that AB = BA = In

Invertible = nonsingular = nondegenerate
Non-Invertible = singular = degenerate
Matrix A is invertible if and only if det 𝑩 ≠ 0

7 October 2019

Is the matrix invertible?

SLIDE 3

M. Hameed

7 October 2019

Invert Z3x3 to determine Z-1

𝒂 = 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 𝑕 ℎ 𝑗 𝒂−1 = 1 det 𝒂 𝑏𝑒𝑘 𝒂 = 1 det 𝒂 𝐵 𝐶 𝐷 𝐸 𝐹 𝐺 𝐻 𝐼 𝐽

𝑈

𝑏𝑒𝑘 𝒂 = Transpose of matrix of cofactors

3×3

④

Transpose

SLIDE 4

M. Hameed

7 October 2019

𝑏𝑒𝑘(𝒂) = 𝐵 𝐶 𝐷 𝐸 𝐹 𝐺 𝐻 𝐼 𝐽

𝑈

𝒂 = 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 𝑕 ℎ 𝑗

𝐵 = 𝑓𝑗 − 𝑔ℎ 𝐶 = − 𝑒𝑗 − 𝑔𝑕 𝐷 = 𝑒ℎ − 𝑓𝑕 𝐸 = − 𝑐𝑗 − 𝑑ℎ 𝐹 = 𝑏𝑗 − 𝑑𝑕 𝐺 = − 𝑏ℎ − 𝑐𝑕 𝐻 = 𝑐𝑔 − 𝑑𝑓 𝐼 = − 𝑏𝑔 − 𝑑𝑒 𝐽 = 𝑏𝑓 − 𝑐𝑒

det 𝒂 = 𝑏𝐵 + 𝑐𝐶 + 𝑑𝐷

SLIDE 5

M. Hameed

7 October 2019 7 October 2019

Example

𝒂 = −1 1 2 1 −1 1 2 −2

' = ?

Firstling

3X 3

adic⇒

= E

=L; ; I;]

den ⇒

÷ ;¥÷

:B

¥,

SLIDE 6

M. Hameed

7 October 2019

Che4

= I