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Determinants and Properties

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Determinant Using Cofactor Expansion

Cofactor expansion calculates the determinant by expanding along a row or column,

\text{det}(A) = \sum_{i=1}^{n} (-1)^{i+j} a_{ij} C_{ij}

, where

C_{ij}

is the cofactor of

a_{ij}

. For

A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & 0 \\ 1 & 0 & 4 \end{pmatrix}

, expanding along the third column,

\text{det}(A) = 0 - 0 + 0(3 \times 0 - 1 \times 0) = 0

Determinant of an Identity Matrix

The determinant of an identity matrix of any size is 1. For example,

\text{det}(I_3)

for the 3x3 identity matrix

I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

is just 1 because the product of the diagonal elements is 1.

Determinant of a Block Matrix

To find the determinant of a block matrix, if the matrix is of the form

\begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}

where A and B are square matrices, then

\text{det}(\text{Block Matrix}) = \text{det}(A) \times \text{det}(B)

. Example: For

\begin{pmatrix} 2 & 0 \\ 0 & \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix} \end{pmatrix}

, the determinant is

2(3 \cdot 2 - 4 \cdot 1) = 4

Laplace Expansion (Cofactor Expansion Theorem)

Laplace Expansion is similar to cofactor expansion and is used for calculating the determinant by developing along a row or column. For instance,

\text{det}\left(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\right) = a \cdot d - b \cdot c

, which is the subtraction of the product of the diagonals.

Row Swapping

Switching two rows of a matrix multiplies its determinant by -1. For example, swapping any two rows of a matrix

A

results in

\text{det}(A') = -\text{det}(A)

. If

A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}

and we swap the rows,

\text{det}(A') = -\text{det}(\begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}) = -(4 - 3) = -1

Row Addition

Adding a multiple of one row to another does not change the determinant. For instance, if we add

\lambda

times row 1 to row 2 in matrix

A

\text{det}(A)

is unchanged. If

A = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}

, and we add 2 times row 1 to row 2, the determinant of

A' = \begin{pmatrix} 1 & 3 \\ 4 & 10 \end{pmatrix}

is still

2 - 6 = -4

Determinant of a Matrix Product

The determinant of a product of two matrices is the product of their determinants,

\text{det}(AB) = \text{det}(A)\times\text{det}(B)

. If

A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}

and

B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}

, then

\text{det}(A) = -2

and

\text{det}(B) = -2

, so

\text{det}(AB) = (-2) \times (-2) = 4

Determinant of a Scalar Multiple

The determinant of a scalar multiple of a matrix is the scalar raised to the power of the dimension, times the determinant of the matrix,

\text{det}(\lambda A) = \lambda^n \text{det}(A)

for an nxn matrix

A

. For example, if

\lambda = 2

and

A = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}

, then

\text{det}(2A) = 2^2(1 \cdot 4 - 3 \cdot 2) = -8

Determinant of a Symmetric Matrix

While there's no special formula for the determinant of a symmetric matrix, it obeys all the standard determinant properties. Given a symmetric matrix

A

A = A^T

, and so

\text{det}(A) = \text{det}(A^T)

. An example is

A = \begin{pmatrix} 2 & 3 \\ 3 & 4 \end{pmatrix}

, where

\text{det}(A) = 2 \cdot 4 - 3 \cdot 3 = -1

Determinant After Row or Column Operations

The determinant changes in specific ways after elementary row or column operations. For example, if a row is multiplied by a non-zero scalar

\lambda

, the determinant is multiplied by

\lambda

. If two rows are swapped, the determinant changes sign. Adding a multiple of one row to another does not change the determinant. Given

A = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}

and multiplying the first row by 3,

\text{det}(A') = 3 \times (3 \cdot 4 - 1 \cdot 2) = 30

Triangular Matrix Determinant

The determinant of a triangular matrix (upper or lower) is the product of its diagonal entries. For an upper triangular matrix

A

\text{det}(A) = a_{11}a_{22}...a_{nn}

. Given

A = \begin{pmatrix} 2 & 1 \\ 0 & 5 \end{pmatrix}

, the determinant is

2 \times 5 = 10

Determinant of the Transpose

The determinant of a matrix and its transpose are the same,

\text{det}(A) = \text{det}(A^T)

. For matrix

A = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}

, both

\text{det}(A)

and

\text{det}(A^T)

equal to

1 \cdot 4 - 3 \cdot 2 = -2

Determinant of a Diagonal Matrix

The determinant of a diagonal matrix is the product of its diagonal entries, just like for triangular matrices. For diagonal matrix

D

\text{det}(D) = d_{11}d_{22}...d_{nn}

. Given

D = \begin{pmatrix} 7 & 0 \\ 0 & 3 \end{pmatrix}

, then

\text{det}(D) = 7 \times 3 = 21

Determinant of a Permutation Matrix

The determinant of a permutation matrix is 1 or -1, depending on whether the number of row swaps from the identity matrix is even (determinant is 1) or odd (determinant is -1). For the permutation matrix

P = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

, which is one swap from the identity, hence

\text{det}(P) = -1

Determinant of a Singular Matrix

A singular (or non-invertible) matrix has a determinant of zero. For example, if matrix

A

has a row of zeros or two proportional rows, then

\text{det}(A) = 0

. Given

A = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}

(second row is twice the first), then

\text{det}(A) = 4 - 4 = 0

Permutations and Determinants

A determinant can be calculated by summing the products of elements and their corresponding signed permutation. For a 2x2 matrix

A

\text{det}(A)

is the sum of the products of the elements of each permutation of the first row with the corresponding elements of the second row, signed by the permutation's parity. For example,

\text{det}\left(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\right) = ad - bc

, using the permutations (a,d) and (b,c).

Row Multiplication

Multiplying a row by a scalar multiplies the determinant by that scalar. For example, if we multiply a row of

A

\lambda

, then

\text{det}(A') = \lambda \cdot \text{det}(A)

. Given

A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}

, multiplying the first row by 2 yields

\text{det}(A') = 2 \cdot \text{det}(\begin{pmatrix} 2 & 4 \\ 3 & 4 \end{pmatrix}) = 2(8 - 12) = -8

Determinant of an Inverse Matrix

The determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix, provided the matrix is invertible,

\text{det}(A^{-1}) = \frac{1}{\text{det}(A)}

. For

A = \begin{pmatrix} 3 & 0 \\ 2 & 1 \end{pmatrix}

with

\text{det}(A) = 3

, the determinant of its inverse is

\text{det}(A^{-1}) = \frac{1}{3}

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