- An element of A is aij
- aij is called the ijth element of A
Multiplication by a Scalar
- Multiply each element of the matrix by the scalar.
- Let c = 3
Matrix Addition & Subtraction
- Add each element of the first matrix to the corresponding element of the second matrix.
- In order to add matrices, the matrices must have the same dimensions.
- Note: A+B = B+A -- commutative law
- Note: A+(B+C) = (A+B)+C -- associative law
- Note: In general, A*B does not equal B*A -- matrix multiplication is not commutative
- Note: (A*B)*C = A*(B*C) -- associative law
- Note: A*(B+C) = A*B + A*C
- And: (B+C)*A = B*A + C*A -- distributive law
Transpose of a Matrix
- Mathematical notation -> X'
- (A')' = A
- (A*B)' = B' * A'
Note the reverse order of the matrix multiplication
- Denoted as u or 1
- Denoted as U or 1
- Note: The number of rows equals the number of columns
- Denoted as I
- Remember I*A = A*I = A
- Remember C = C'
Inverse of a Matrix
Inversion by Hand
Let's find the inverse of this matix:
First we will augment Matrix A with an identity matrix and call the result A0.
We will also need an identity matrix called I0.
Now we will work on the first column of I0. The diagonal element in the first column of
A0 is called the pivot. We will replace the first diagonal element of I0
with 1/a11. Then we will
replace the non-pivot elements with -aij/a11 and call the result I1.
Compute A1 = I1 * A0.
Now we repeat the process by modifying the second column of I0.
This time the pivot value is the 22 element. The new matrix is called I2.
Compute A2 = I2 * A1.
Lastly we repeat the process by modifying the third column of I0.
This time the pivot value is the 33 element. The new matrix is called I3.
Compute A3 = I3 * A2.
The last three columns of A3 are the inverse of A, thus:
a = (4,2,2\4,6,8\-2,2,4)
a = a,i
i1 = (.25,0,0\-1,1,0\.5,0,1)
i2 = (1,-.125,0\0,.25,0\0,-.75,1)
ainv = i3*i2*i1*a
ainv = ainv[1..., 4..6]
Determinant of a Matrix
- The fundamental geometric meaning of the determinant is as the scale factor
for volume when matrix A is regarded as a linear transformation.
- Mathematical notation: d = |A|
Example: Inverse & Determinant of a Matrix
Computing Column & Row Sums
Vertical Concatenation (Appending)
Multivariate Course Page
Phil Ender, 11oct05, 30Jun98