Properties of the determinants of matrices

The determinant of a matrix is a scalarA numeric value. value that helps us understand a matrix's behavior and properties. It is only applicable for a square matrix. For a matrix $M$, we can represent its determinant as $det(M)$ or $|M|$.

Note: To learn more about a determinant and he can we calculate it, we refer to this Answer.

Properties of determinants find application in various fields of mathematics to help us perform matrix operations and solve linear equations. In this Answer, we will be discussing various properties of determinants, including the following.

Note: We suppose all the matrices we discuss below are square matrices.

1) Transpose property

If we perform the transposeInterchanging rows with columns or vice versa. of a matrix, the result of the determinant is unchanged. For a matrix $M$, we can represent this as:

Note: The transpose property is also known as the reflection property.

Example

Suppose we have a matrix $M$ such that:

$M = \begin{bmatrix}1 & 2 \\3 & 4 \\\end{bmatrix}$

$M^T = \begin{bmatrix}1 & 3 \\2 & 4 \\\end{bmatrix}$

• Calculating $det(M)$:

• Calculating $det(M^T)$:

We conclude $det(M) = det(M^T)$.

2) All zero rows/columns property

If a matrix contains a zero row or zero column (having all zero values), then the determinant of the matrix is equal to $0$.

Example

Suppose we have a matrix $M$ such that:

$M = \begin{bmatrix}0 & 0 \\3 & 4 \\\end{bmatrix}$

Here the 1st row consists of all zero values, so $det(M)$ should be equal to zero.

• Calculating $det(M)$:

We conclude $det(M)=0$ when a row or a column in a matrix consists of all zero values.

3) Interchanging rows/columns property

When we interchange two rows or columns in a matrix, its determinant is multiplied by $-1$, meaning the determinant's sign is reversed. For a matrix $M$, if we interchange two rows or columns, we can represent this as:

Example

Suppose we have a matrix $M$ such that:

$M = \begin{bmatrix}1 & 2 \\3 & 4 \\\end{bmatrix}$

• Calculating $det(M)$:

• Interchanging the 1st and 2nd row, so the matrix $M$becomes:
  $M = \begin{bmatrix}3 & 4 \\1 & 2\\\end{bmatrix}$

• Calculating $det(M)$:

We conclude $det(M)=-det(M)$ when we interchange two rows or columns in a matrix.

4) Identity matrix property

An identity matrix is a matrix in which the main diagonal consists of all $1$ elements and the rest elements are $0$ . For an identity matrix, the determinant equals 1.

Example

Suppose we have an identity matrix $M$ such that:

$M = \begin{bmatrix}1 & 0 \\0 & 1 \\\end{bmatrix}$

• Calculating $det(M)$:

We conclude $det(M)=0$ for an identity matrix.

5) Identical rows/columns property

If we have two same rows or columns in a matrix, meaning all the elements are identical, the determinant equals 0.

Example

Suppose we have a matrix $M$ such that:

$M = \begin{bmatrix}3 & 4 \\3 & 4 \\\end{bmatrix}$

Here the 1st and 2nd rows are identical.

• Calculating $det(M)$:

We conclude $det(M)=0$ when we have two same rows or columns in a matrix.

6) Scalar multiplication property

If we multiply a scalar with all elements of a row or column of a matrix, the determinant is multiplied by the same scalar. For a matrix $M$, we represent this as:

Here $a$ is a scalar.

Example

Suppose we have a matrix $M$ such that:

$M = \begin{bmatrix}1 & 2 \\3 & 4 \\\end{bmatrix}$

• Calculating $det(M)$:

• Multiplying 1st row with $2$, now $M$ becomes:

$M = \begin{bmatrix}2 & 4 \\3& 4 \\\end{bmatrix}$

• Calculating $det(M)$:

We conclude $det(M) = 2×det(M)$ when a row or column of a matrix is multiplied by the scalar $2$.

7) Sum property

If a row or column of a matrix consists of the sum of two values, then the determinant of this matrix equals the determinant of two matrices breaking these elements. We represent this as:

Here $M = \begin{bmatrix}a1+a2 & b \\c1+c3 & d \\\end{bmatrix}$ , $N= \begin{bmatrix}a1 & b \\c1 & d \\\end{bmatrix}$ and$P= \begin{bmatrix}a2 & b \\c2 & d \\\end{bmatrix}$.

Example

Suppose we have a matrix $M$ such that:

$M = \begin{bmatrix}1+5 & 2 \\3+6 & 4 \\\end{bmatrix}$

• Calculating $det(M)$:

• We split matrix $M$ into matrix $N$ and $P$ such that:

$N = \begin{bmatrix}1 & 2 \\3 & 4 \\\end{bmatrix}$

$P = \begin{bmatrix}5 & 2 \\6 & 4 \\\end{bmatrix}$

• Calculating $det(N)$:

• Calculating $det(P)$:

We conclude $det(M)=det(N) +det(P)$.

8) Triangular property

A triangular matrix is a matrix in which either the elements above the main diagonal are $0$ or the elements below the main diagonal are $0$ . The matrix is still triangular if both conditions are true. The determinant of a triangular matrix equals the product of the main diagonal elements. We represent this as:

Here $a$ and $d$ are scalars and$M= \begin{bmatrix}a & 0 \\c & d \\\end{bmatrix}$ or $M= \begin{bmatrix}a & b \\0 & d \\\end{bmatrix}$ or $M= \begin{bmatrix}a & 0 \\0 & d \\\end{bmatrix}$

Example

Suppose we have a matrix $M$ such that:

$M = \begin{bmatrix}1 & 0 \\3 & 4 \\\end{bmatrix}$

• Calculating $det(M)$:

• Calculating product of main diagonal elements.

We conclude $det(M)$ is equal to product of main diagonal values for a triangular matrix.

9) Factor property

If a determinant becomes $0$ when we substitute $x$ with a value $k$, then $(x – k)$ is a factor of that determinant.

Example

Suppose we have a matrix $M$ such that:

$M = \begin{bmatrix}x & 1 \\4 & x \\\end{bmatrix}$

• Calculating $det(M)$:

• We substitute the value of $x=2$:

We conclude $(x-2)$ is a factor of $det(M)$.

10) Co-factor property

If we take the co-factor of a matrix with order $n$ then the resultant determinant of this co-factor matrix will be equal to $(n-1)th$ power of the determinant of the original matrix. We represent this as:

Here $C$ is the the co-factor matrix of $M$.

Example

Suppose we have a matrix $M$with order 2 such that:

$M = \begin{bmatrix}1 & 2 \\3 & 4 \\\end{bmatrix}$

• Calculating $det(M)$:

• Now we calculate the co-factor matrix of $M$ represent as $C$:
  $C = \begin{bmatrix}4 & -3 \\-2 & 1 \\\end{bmatrix}$

• Calculating $det(C)$:

We conclude $det(C)$is equal to $(det(M))^1$ for a matrix with order 2.

Conclusion

The properties of determinants play a pivotal role in simplifying matrix operations and understanding the behavior of matrices in various mathematical contexts. These properties help us in matrix transformations, equations, and geometric interpretations.