MIT 18.06 Lecture 18: Properties of Determinants

linear algebra

MIT 18.06

determinants

matrix properties

Ten fundamental properties that completely define the determinant and reveal when matrices are invertible

Author

Chao Ma

Published

November 3, 2025

Every square matrix has a number called the determinant.

\[ \det A=|A| \]

The matrix is invertible when the determinant is not zero
The matrix is singular when the determinant is zero

The Ten Properties

1. Identity Matrix

\[ \det I=1 \]

2. Row Exchange

Exchanging rows changes the sign of the determinant.

Determinant of permutation matrices is: - 1, when number of exchanges is even - -1, when number of exchanges is odd

\[ \begin{vmatrix}1&0\\0&1\end{vmatrix}=1 \]

\[ \begin{vmatrix}0&1\\1&0\end{vmatrix}=-1 \]

3. Linearity in Each Row

The determinant is linear in each row separately.

\[ \begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc \]

\[ \begin{vmatrix}ta&tb\\c&d\end{vmatrix}= t\begin{vmatrix}a&b\\c&d\end{vmatrix} \tag{3a} \]

\[ \begin{vmatrix}a+a'&b+b'\\c&d\end{vmatrix}=\begin{vmatrix}a&b\\c&d\end{vmatrix}+ \begin{vmatrix}a'&b'\\c&d\end{vmatrix} \tag{3b} \]

4. Equal Rows

Two equal rows lead to determinant 0.

Proof: If two rows are equal, exchanging them doesn’t change the matrix. But by property 2, exchanging rows changes the sign of the determinant. Therefore $\det A = -\det A$, which means $\det A = 0$.

5. Row Operations

Subtracting $l \times \text{row}_i$ from $\text{row}_k$ doesn’t change the determinant.

\[ \begin{vmatrix}a&b\\c-la&d-lb\end{vmatrix}=\begin{vmatrix}a&b\\c&d\end{vmatrix}+\begin{vmatrix}a&b\\-la&-lb\end{vmatrix}=\\ \begin{vmatrix}a&b\\c&d\end{vmatrix}-l\begin{vmatrix}a&b\\a&b\end{vmatrix} \]

The last term is zero by property 4 (equal rows), so the determinant remains unchanged.

6. Zero Rows

Rows of zeros lead to determinant 0.

\[ 5\begin{vmatrix}0&0\\c&d\end{vmatrix}=\begin{vmatrix}5 \times 0&5 \times 0\\c&d\end{vmatrix}= \begin{vmatrix}0&0\\c&d\end{vmatrix} \]

If $5 \times n=n$, then $n=0$.

7. Triangular Matrix

The determinant of a triangular matrix is the product of the diagonal elements (pivots).

\[ |U|= \begin{vmatrix}d_1&*&*&*\\0&d_2&*&*\\0&0&d_3&*\\0&0&0&d_4\end{vmatrix}=d_1 \times d_2 \times ... \times d_n \]

From property 5, we can change an upper triangular matrix to a diagonal matrix with the same determinant. Then using property 3a, we can factor out each diagonal element:

\[ |U|=d_1 \times d_2 \times ... \times d_n\begin{vmatrix}1&0&0&0\\0&1&0&0\\0&0&\ddots&0\\0&0&0&1\end{vmatrix}=d_1 \times d_2 \times ... \times d_n \]

8. Singular Matrix

The determinant is 0 exactly when A is singular.

When the determinant is not 0, A is invertible.

9. Product of Matrices

\[ \det AB=(\det A)(\det B) \]

This property leads to:

\[ \det A^{-1}=\frac{1}{\det A} \]

\[ \det A^2=(\det A)^2 \]

\[ \det 2A=2^n (\det A) \]

10. Transpose

\[ \det A^\top=\det A \]

Proof:

\[ |A|=|LU| \]

\[ |A^\top|=|U^\top L^\top| \]

Because both $L$ and $U$ are triangular matrices, the determinant is just the product of diagonal elements, so the transpose doesn’t change the result.

Source: MIT 18.06SC Linear Algebra, Lecture 18

--- title: "MIT 18.06 Lecture 18: Properties of Determinants" author: "Chao Ma" date: "2025-11-03" categories: [linear algebra, MIT 18.06, determinants, matrix properties] description: "Ten fundamental properties that completely define the determinant and reveal when matrices are invertible" --- {{< video https://www.youtube.com/watch?v=srxexLishgY >}} --- Every square matrix has a number called the determinant. $$ \det A=|A| $$ - The matrix is **invertible** when the determinant is not zero - The matrix is **singular** when the determinant is zero --- ## The Ten Properties ### 1. Identity Matrix $$ \det I=1 $$ ### 2. Row Exchange Exchanging rows changes the sign of the determinant. Determinant of permutation matrices is: - **1**, when number of exchanges is even - **-1**, when number of exchanges is odd $$ \begin{vmatrix}1&0\\0&1\end{vmatrix}=1 $$ $$ \begin{vmatrix}0&1\\1&0\end{vmatrix}=-1 $$ ### 3. Linearity in Each Row The determinant is linear in each row separately. $$ \begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc $$ $$ \begin{vmatrix}ta&tb\\c&d\end{vmatrix}= t\begin{vmatrix}a&b\\c&d\end{vmatrix} \tag{3a} $$ $$ \begin{vmatrix}a+a'&b+b'\\c&d\end{vmatrix}=\begin{vmatrix}a&b\\c&d\end{vmatrix}+ \begin{vmatrix}a'&b'\\c&d\end{vmatrix} \tag{3b} $$ ### 4. Equal Rows Two equal rows lead to determinant 0. **Proof**: If two rows are equal, exchanging them doesn't change the matrix. But by property 2, exchanging rows changes the sign of the determinant. Therefore $\det A = -\det A$, which means $\det A = 0$. ### 5. Row Operations Subtracting $l \times \text{row}_i$ from $\text{row}_k$ doesn't change the determinant. $$ \begin{vmatrix}a&b\\c-la&d-lb\end{vmatrix}=\begin{vmatrix}a&b\\c&d\end{vmatrix}+\begin{vmatrix}a&b\\-la&-lb\end{vmatrix}=\\ \begin{vmatrix}a&b\\c&d\end{vmatrix}-l\begin{vmatrix}a&b\\a&b\end{vmatrix} $$ The last term is zero by property 4 (equal rows), so the determinant remains unchanged. ### 6. Zero Rows Rows of zeros lead to determinant 0. $$ 5\begin{vmatrix}0&0\\c&d\end{vmatrix}=\begin{vmatrix}5 \times 0&5 \times 0\\c&d\end{vmatrix}= \begin{vmatrix}0&0\\c&d\end{vmatrix} $$ If $5 \times n=n$, then $n=0$. ### 7. Triangular Matrix The determinant of a triangular matrix is the **product of the diagonal elements (pivots)**. $$ |U|= \begin{vmatrix}d_1&*&*&*\\0&d_2&*&*\\0&0&d_3&*\\0&0&0&d_4\end{vmatrix}=d_1 \times d_2 \times ... \times d_n $$ From property 5, we can change an upper triangular matrix to a diagonal matrix with the same determinant. Then using property 3a, we can factor out each diagonal element: $$ |U|=d_1 \times d_2 \times ... \times d_n\begin{vmatrix}1&0&0&0\\0&1&0&0\\0&0&\ddots&0\\0&0&0&1\end{vmatrix}=d_1 \times d_2 \times ... \times d_n $$ ### 8. Singular Matrix The determinant is 0 **exactly when A is singular**. When the determinant is not 0, A is invertible. ### 9. Product of Matrices $$ \det AB=(\det A)(\det B) $$ This property leads to: $$ \det A^{-1}=\frac{1}{\det A} $$ $$ \det A^2=(\det A)^2 $$ $$ \det 2A=2^n (\det A) $$ ### 10. Transpose $$ \det A^\top=\det A $$ **Proof**: $$ |A|=|LU| $$ $$ |A^\top|=|U^\top L^\top| $$ Because both $L$ and $U$ are triangular matrices, the determinant is just the product of diagonal elements, so the transpose doesn't change the result. --- *Source: MIT 18.06SC Linear Algebra, Lecture 18*