What are Determinants
Determinants by Cofactor Expansion
Determinants by Row Reduction
Cramer's Rule
A determinant is a mapping from a matrix to a single number
\[ \det(A) = d, \text{where}\ d \in \mathbb{R} \]
It can be represented by det() or by |, as below:
\[ \det(A) = \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix}\]
Determinants are only defined for square matrices
They convey important information about the matrix:
A matrix is singular if and only if its determinant is zero
For two by two matrices, the determinant is equal to the area of the parallelogram defined by its columns
For a general 1 X 1 Matrix
\[ A_{1,1} = \begin{bmatrix} a_{11}\\ \end{bmatrix}\]
Its determinant is \[ \det(A_{1,1}) = a_{11} \]
For a general 2 X 2 Matrix
\[ A_{2,2} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{21} \\ \end{bmatrix}\]
Its determinant is calculated as: \[ \det(A_{2,2}) = a_{11}a_{22} - a_{12}a_{21}\]
Combining both results, we can show that
\[ \det(A_{2,2}) = \det(a_{11})det(a_{22}) - \det(a_{12})\det(a_{21})\]
Mij is the determinant of the submatrix after the ith row and jth column are removed from A
Mij is known as the minor of entry aij
Cij = (-1)i+jMij is known as the cofactor of entry aij
\[A = \begin{bmatrix} 3 & 2 & 4\\ 2 & 1 & 3\\ 1 & 2 & 2\\ \end{bmatrix} \]
The determinant of an n X n matrix can be calculated using cofactors.
Across a column:
\[ \det(A_{n,n}) = a_{1j}C_{1j} + a_{2j}C_{2j} + \cdots + a_{nj}C_{nj}\]
Across a row:
\[ \det(A_{n,n}) = a_{j1}C_{j1} + a_{j2}C_{j2} + \cdots + a_{jn}C_{jn}\]
\[A = \begin{bmatrix} 3 & 2 & 4\\ 2 & 1 & 3\\ 1 & 2 & 2\\ \end{bmatrix} \]
For 3 X 3 matrices, the determinant is obtained by:
→ Extending the first columns rows to the right
→ Add the product of the three left-to-right downward diagonals
→ Add the product of the three right-to-left downward diagonals
→ Subtract the first result by the second
\[A = \begin{bmatrix} 3 & 2 & 4 & \vdots & 3 & 2\\ 2 & 1 & 3 & \vdots & 2 & 1\\ 1 & 2 & 2 & \vdots & 1 & 2\\ \end{bmatrix} \]
If A has a row or columns of zeros, then det(A) = 0
If B is obtained by multiplying a row or column of a matrix A by k, det(B) = k det(A)
If B is obtained by swapping two rows or two columns of a matrix A, then det(B) = - det(A)
If B obtained by adding a multiple of a row or column of a matrix A into another, then det(B) = det(A)
If a matrix has two or more proportional rows or columns, det(A) = 0
\[A = \begin{bmatrix} 3 & 0 \\ 2 & 0 \\ \end{bmatrix}, B = \begin{bmatrix} 2 & 1 \\ 0 & 2 \\ \end{bmatrix} \]
\[C = \begin{bmatrix} 2 & 1 \\ 0 & 1 \\ \end{bmatrix}, D = \begin{bmatrix} 2 & 0 \\ 0 & 1 \\ \end{bmatrix} \]
These properties can be used to help calculate determinants:
→ Take steps to put the matrix in upper triangularform
→ Keep track of how the determinant of this new matrix compares with the original one
→ Calculate the determinant of the upper triangular matrix, and then obtain the determinant of the original matrix
\[A = \begin{bmatrix} 3 & 2 & 4\\ 2 & 1 & 3\\ 1 & 2 & 2\\ \end{bmatrix} \]
If E is an elementary matrix, det(EA) = det(E)det(A)
A square matrix A is invertible if and only if det(A) ≠ 0
From the two previous properties, one can show that: det(AB) = det(A)det(A)
From the previous properties, one can show that: det(A-1) = 1/det(A)
If Ax = b is a system of equations and det(A) ≠ 0, then
→ The system has a unique solution
→ It's solution is given by:
\[ x_{1} = \frac{\det(A_{1})}{\det(A)}, x_{2} = \frac{\det(A_{2})}{\det(A)}, \ldots, x_{1} = \frac{\det(A_{n})}{\det(A)} \]
Where Aj is the matrix obtained by replacing the jjth column of A with b
Let's solve:
\[\begin{aligned} x + y + z & = 6 \\ x + 2y + 2z & = 11 \\ 2x + 3y - z & = 5 \end{aligned} \]
Using Cramer's rule
The following statements are equivalent:
(a) A is invertible
(b) Ax = 0 has only the trivial solution
(c) The reduced form echelon of A is I
(d) A can be expressed as a product of Ek{k}
(e) Ax = b has only one solution for every matrix b/
(f) The determinant of A is non-zero
Textbook: Sections 2.1, 2.2, 2.3
Exercises
→ Section 2.1 1, 5, 6, 7, 10, 11, 12, 19a, 19b
→ Section 2.2 5, 6, 7, 8, 9, 10, 11, 12
→ Section 2.2 24, 25, 26