Determinants and Inverses :
Determinants and Inverses How do you calculate determinants and inverses of 2 x 2 and 3 x 3 matrices? Slide 2:
A multiplicative identity matrix is an n x n matrix with 1s
along the main diagonal and 0s elsewhere. 2 x 2 Multiplicative Identity 3 x 3 Multiplicative Identity Slide 3:
Remember from a previous lesson: If ab = 1, then b
is the multiplicative inverse of a so… If A and X are square matrices and AX = XA = I,
where I is the identity matrix, then X is the multiplicative
inverse of A–written A-1. Slide 4:
To show that 2 matrices are inverses of each other,
show AB = I or BA = I. For example, show that B is the multiplicative inverse of A: Show that AB = I. Slide 5:
Every square matrix with real number elements has
a real number determinant. Determinants help you
find inverses. To find the determinant of a 2 x 2 matrix, use the
following formula: Notice: The brackets change to lines to represent
Determinants. Slide 6:
For example: 6(-3) – 2(-1) =-18 + 2
=-16 6(2) – 1(-3) =12 + 3
=15 a(a) – 3(2) =a2 - 6 Slide 7:
To find the determinants of a 3 x 3 matrix, you will first
need to recopy the first two columns. 1. Multiply the downward diagonals.
2. Multiply the upward diagonals.
3. Add the results.
4. Subtract the top sum from the bottom sum. Slide 8:
Find the determinant of the matrix below: 1. Multiply the upward diagonals. We get 2, 0, 0. 2. Multiply the downward diagonals. We get 6, -12, and 0. 3. Find the sums. Sum = 2 Sum = -6 4. Subtract the top sum from the bottom sum. Det = -6 – 2 = -8!! Slide 9:
Another example:
Find the determinant of Recopy the first
2 columns. Multiply the upward diagonals.
Multiply the downward diagonals.
Find the sums.
Subtract the sums. We get -9, 16, and 0. We get 2, -6, and 0. Sum = 7 Sum = -8 Determinant = -8 – 7 = -15!!