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Linear Algebra - review problems
1 Determinants
Let A be a n × n square matrix with coefficients aij (1 ≤ i, j ≤ n). The determinant of A is computed
as follows:
|A| =
∑^ n
j=
ajα (−1)
j+α | A˜ (^) j α| (1)
where α is a fixed integer between 1 and n and A˜ (^) jα is the (n − 1) × (n − 1) matrix obtained from A by
removing the j-th row and the α-th column. A˜ (^) jα has coefficients:
A˜
jα =
a 11... a 1 ,(α−1) a 1 ,(α+1)... a 1 n
. . .
a(j−1), 1... a(j−1),(α−1) a(j−1),(α+1)... a(j−1),n
a(j+1), 1... a(j+1),(α−1) a(j+1),(α+1)... a(j+1),n
. . .
an 1... an,(α−1) an,(α+1)... an,n
Note that column α and row j are missing.
Some properties of the determinant:
1. |AB| = |A| · |B|,
- if |A| $= 0, then A is non singular and A
− 1 exists. Then |A
− 1 | = 1/|A|.
- if |A| $= 0, then the columns of A are linearly independent. Defining the column vector a (^) k as the
k-th column of A – that means that a (^) k =
a 1 k
a 2 k
. . .
ank
A · x =
∑^ n
j=
xj a (^) j = 0 ⇔ x = 0.
If a linear combination of columns of A is equal to zero, the coefficients (xi ) of the combination are
all equal to zero.
- Conversely, |A| $= 0 is equivalent to any of the three following statements:
− there exists a linear combination of columns of A equal to zero with coefficients that are not
all equal to zero:
∑n
j=
xj aj = 0 with at least one non zero xj.
− there exists x $= 0 such that A · x = 0,
− one of the column of A can be expressed as a linear combination of the others.
(Make sure that you understand why these three statements are equivalent).
- Exchanging two rows or columns of a matrix flips the sign of the determinant.
- Adding to one row the product of another row by a constant λ lives the determinant unchanged.
Example: ∣ ∣ ∣ ∣
a b
c d
a b
c + λa d + λb
In the same way, adding to a column of A the product of another column with a constant μ lives
the determinant unchanged. Example:
a b
c d
a b + μa
c d + μc
- The determinant of a diagonal matrix is equal to the product of the diagonal elements.
- The determinant of an upper or lower triangular matrix is equal to the product of its diagonal
elements.
- The determinant of a 2 × 2 matrix A is computed as follows:
|A| =
a b
c d
∣ =^ ad^ −^ bc.
Problems
Compute the determinant of the following matrices:
1. A =
2. B =
3. C =
with D =
λ (^1)
.. .
λ (^) n
the diagonal matrix with diagonal coefficients equal to the eigenvalues of
A, and V a matrix whose columns are eigenvectors v (^) i associated with each of the different eigenvalues.
The first column of V is v 1 , an eigenvector associated with λ 1 , the second column is v 2 , an eigenvector
associated with λ 2 , etc....
Problems
Find the eigendecomposition of the following matrices:
1. A 1 =
2. A 2 =
