Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Inverse & Identity MATRICES Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: October 12, 2005
Inverses and Identities 5x = 3 Jeff Bivin -- LZHS
Now with Matrices ? This is the Identity Matrix for 2 x 2 Matrices Jeff Bivin -- LZHS
Let’s look at another example Jeff Bivin -- LZHS
What do we multiply a matrix by to get the Identity? New Question ? What do we multiply a matrix by to get the Identity? Jeff Bivin -- LZHS
The Inverse of a 2x2 Matrix Jeff Bivin -- LZHS
The Inverse of a 2x2 Matrix Jeff Bivin -- LZHS
The Inverse of a 2x2 Matrix Jeff Bivin -- LZHS
A B This is our Formula! Jeff Bivin -- LZHS
A B This is our Formula! Jeff Bivin -- LZHS
A B C This is our Formula! Jeff Bivin -- LZHS
Are the two Matrices Inverses? The product of inverse matrices is the identity matrix. Identity, therefore, INVERSE Matrices Jeff Bivin -- LZHS
Are the two Matrices Inverses? The product of inverse matrices is the identity matrix. Not the Identity, therefore, NOT INVERSE Matrices Jeff Bivin -- LZHS
Does the Matrix have an Inverse? Let’s review the definition of the Inverse of a 2x2 Matrix Jeff Bivin -- LZHS
The Inverse of a 2x2 Matrix Jeff Bivin -- LZHS
Does the Matrix have an Inverse? Find the determinant! Therefore, NO inverse! Jeff Bivin -- LZHS
Does the Matrix have an Inverse? Find the determinant! Therefore, an inverse exists! Jeff Bivin -- LZHS
Does the Matrix have an Inverse? + + 1 2 3 4 5 6 7 8 9 Find the determinant! - - - 1•5•9 + 2•6•7 + 3•4•8 - 7•5•3 - 8•6•1 - 9•4•2 45 + 84 + 96 - 105 - 48 - 72 Therefore, NO inverse! Jeff Bivin -- LZHS
Does the Matrix have an Inverse? + + 1 3 2 4 Find the determinant! - - - 1•4•2 + 3•1•3 + 2•2•4 - 3•4•2 - 4•1•1 - 2•2•3 8 + 9 + 16 - 24 - 4 - 12 -7 Therefore, an inverse exists! Jeff Bivin -- LZHS
Solve the system using inverse matrices B 3x + 2y = 7 4x - 5y = 11 This is our Formula! Jeff Bivin -- LZHS
Solve the system using inverse matrices B 2x - 4y = 9 3x - 2y = 1 This is our Formula! Jeff Bivin -- LZHS