Does the determinant of a triangular matrix depend on size?

A determinant of an upper (or lower) triangular matrix is just its diagonal elements product. This holds for any size, even large ones.

Can the determinant be used to solve systems of equations?

Yes, determinants can be used in Cramer's rule to solve systems of linear equations. The system has a unique solution if and only if the determinant of the coefficient matrix is non-zero.

What happens if two rows or columns of a matrix are identical?

In this case, the determinant is zero. This happens whenever the matrix is singular with less than full rank.

How do we compute a larger determinant?

The determinant of a bigger matrix, say 3×3, can be obtained using cofactor expansion. For this expansion, we expand along one chosen row or column.

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Properties of Determinants

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Properties of Determinants

A determinant is a special number that can be found in a square matrix. Determinants carry important information regarding the matrix and play key roles in many areas of mathematics, especially in linear algebra, calculus, and systems of linear equations.

1.0Determinants of a Square Matrix

The determinant is the scalar quantity attached to each square matrix that yields information about the type of matrix, namely whether it is invertible or not.

2.0Properties of Determinants

Property 1: Determinant of a Diagonal Matrix

The determinant is the multiplication or the product of the diagonal elements. For example, given a diagonal matrix $A = d ia g (a_{1}, a_{2}, ...., a_{n}),$

$d e t (A) = a_{1} a_{2} ...... a_{n}$

Property 2: Determinant of the Identity Matrix

The determinant of any identity matrix I_n of any order is always equals 1:

$d e t (I_{n}) = 1$

Property 3: Determinant of a Triangular Matrix

In a triangular matrix, whether it is upper or lower triangular, the determinant of such matrix always equals the product of its diagonal elements:

det(A)=Product of diagonal elements of A

Property 4: Swapping of two Rows/Columns

In a matrix, if two rows/columns are swapped, the sign of determinant changes:

det(A')=-det(A)

Property 5: Multiplying a Row/Column by a Scalar

When a matrix is multiplied by a scalar, say k, the determinant is also multiplied by k:

$d e t (k A) = k . d e t (A)$

Property 6: Adding a multiple of One Row/Column to Another Row/Column

If one row (or column) of a matrix is replaced by the sum of itself and a scalar multiple of another row (or column), then the determinant is unchanged:

det(A')=det(A)

Property 7: Determinant of an Inverse Matrix

The inverse of a matrix implies that a matrix, say A, can be invertible, such that its product is always equal to 1.

$d e t (A^{- 1}) = \frac{1}{d e t ( A )}$

Property 8: Determinant of a Product of Matrices

The determinant of a product of matrices is defined for two square matrices, A and B, of equal size: the determinant of their product is equal to the product of individual determinants:

det(AB)=det(A).det(B)

Property 9: Determinant of a Matrix with a Row or Column of Zeroes

A matrix has a zero determinant if elements of two rows(or columns) are equal, the matrix is singular and lacks an inverse; its determinant is zero.

det(A) =0

Property 10: Determinant of a Matrix with a Row or Column of Zeroes

The determinant of any matrix whose row or column is zero will always be zero. This is because a matrix with a row or column consisting wholly of zeros is singular and does not have a unique solution to the associated system of equations.

det(A) = 0

To get a better understanding of the topic, let’s explore the properties of determinants with examples that are useful in performing important mathematical operations on determinants.

3.0Solved Examples

Problem 1: Find the determinant of the matrix:

A = $247358169$

Solution: Let’s expand the determinant along the first row:

$d e t (A) = 2 \times 5869 - 3 \times 4769 + 1 \times 4758$

$d e t (A) = 2 (5 \times 9 - 8 \times 6) - 3 (4 \times 9 - 7 \times 6) + 1 (8 \times 4 - 5 \times 7)$

$d e t (A) = 2 (45 - 48) - 3 (36 - 42) + (32 - 35) = 2 (- 3) - 3 (- 6) + (- 3)$

$d e t (A) = - 6 + 18 - 3 = 18 - 9 = 9$

Problem 2: Find the determinant of the matrix:

B = $210431652$

Solution: Performing row operation to simplify the given matrix

Subtract the 2 multiplied by the first row from the second to get a new second row:

$R_{2} \to R_{1} - 2 R_{2}$ ,

$2 - 1 0 4 - 1 1 6 - 1 2$

Now, add the first row to the third row to get a new third row:

$R_{3} \to R_{3} + R_{1}$

$2 - 1 2 4 - 1 5 6 - 1 8$

Now, calculate the determinant by expanding along the first row:

$d e t (B) = 2 - 1 5 - 1 8 - 4 - 1 2 - 1 8 + 6 - 1 2 - 1 5$

$d e t (B) = 2 (8 \times - 1 - 5 \times - 1) - 4 (8 \times - 1 - 2 \times - 1) + 6 (5 \times - 1 - 2 \times - 1)$

$d e t (B) = 2 (- 3) - 4 (- 5) + 6 (- 3)$

$d e t (B) = - 6 + 20 - 18 = 0$

Problem 3: Find the determinant of the matrix:

C = $321132243$

Solution: Here, we will perform a column operation to simplify the matrix:

Subtract 2 multiplied by the first column from the second column:

$C_{2} \to C_{2} - 2 C_{1}$

$321 - 5 - 3 0 243$

Calculate the simplified matrix along the first row

$d e t (C) = 3 \times - 3 0 43 - (- 5) \times 2143 + 2 \times 21 - 3 0$

$d e t (C) = 3 (- 9 - 0) + 5 (6 - 4) + 2 (0 + 3)$

$d e t (C) = - 27 + 10 + 6 = - 27 + 16 = - 11$

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Properties of Determinants

A determinant is a special number that can be found in a square matrix. Determinants carry important information regarding the matrix and play key roles in many areas of mathematics, especially in linear algebra, calculus, and systems of linear equations.

1.0Determinants of a Square Matrix

The determinant is the scalar quantity attached to each square matrix that yields information about the type of matrix, namely whether it is invertible or not.

2.0Properties of Determinants

Property 1: Determinant of a Diagonal Matrix

The determinant is the multiplication or the product of the diagonal elements. For example, given a diagonal matrix $A = d ia g (a_{1}, a_{2}, ...., a_{n}),$

$d e t (A) = a_{1} a_{2} ...... a_{n}$

Property 2: Determinant of the Identity Matrix

The determinant of any identity matrix I_n of any order is always equals 1:

$d e t (I_{n}) = 1$

Property 3: Determinant of a Triangular Matrix

In a triangular matrix, whether it is upper or lower triangular, the determinant of such matrix always equals the product of its diagonal elements:

det(A)=Product of diagonal elements of A

Property 4: Swapping of two Rows/Columns

In a matrix, if two rows/columns are swapped, the sign of determinant changes:

det(A')=-det(A)

Property 5: Multiplying a Row/Column by a Scalar

When a matrix is multiplied by a scalar, say k, the determinant is also multiplied by k:

$d e t (k A) = k . d e t (A)$

Property 6: Adding a multiple of One Row/Column to Another Row/Column

If one row (or column) of a matrix is replaced by the sum of itself and a scalar multiple of another row (or column), then the determinant is unchanged:

det(A')=det(A)

Property 7: Determinant of an Inverse Matrix

The inverse of a matrix implies that a matrix, say A, can be invertible, such that its product is always equal to 1.

$d e t (A^{- 1}) = \frac{1}{d e t ( A )}$

Property 8: Determinant of a Product of Matrices

The determinant of a product of matrices is defined for two square matrices, A and B, of equal size: the determinant of their product is equal to the product of individual determinants:

det(AB)=det(A).det(B)

Property 9: Determinant of a Matrix with a Row or Column of Zeroes

A matrix has a zero determinant if elements of two rows(or columns) are equal, the matrix is singular and lacks an inverse; its determinant is zero.

det(A) =0

Property 10: Determinant of a Matrix with a Row or Column of Zeroes

The determinant of any matrix whose row or column is zero will always be zero. This is because a matrix with a row or column consisting wholly of zeros is singular and does not have a unique solution to the associated system of equations.

det(A) = 0

To get a better understanding of the topic, let’s explore the properties of determinants with examples that are useful in performing important mathematical operations on determinants.

3.0Solved Examples

Problem 1: Find the determinant of the matrix:

A = $247358169$

Solution: Let’s expand the determinant along the first row:

$d e t (A) = 2 \times 5869 - 3 \times 4769 + 1 \times 4758$

$d e t (A) = 2 (5 \times 9 - 8 \times 6) - 3 (4 \times 9 - 7 \times 6) + 1 (8 \times 4 - 5 \times 7)$

$d e t (A) = 2 (45 - 48) - 3 (36 - 42) + (32 - 35) = 2 (- 3) - 3 (- 6) + (- 3)$

$d e t (A) = - 6 + 18 - 3 = 18 - 9 = 9$

Problem 2: Find the determinant of the matrix:

B = $210431652$

Solution: Performing row operation to simplify the given matrix

Subtract the 2 multiplied by the first row from the second to get a new second row:

$R_{2} \to R_{1} - 2 R_{2}$ ,

$2 - 1 0 4 - 1 1 6 - 1 2$

Now, add the first row to the third row to get a new third row:

$R_{3} \to R_{3} + R_{1}$

$2 - 1 2 4 - 1 5 6 - 1 8$

Now, calculate the determinant by expanding along the first row:

$d e t (B) = 2 - 1 5 - 1 8 - 4 - 1 2 - 1 8 + 6 - 1 2 - 1 5$

$d e t (B) = 2 (8 \times - 1 - 5 \times - 1) - 4 (8 \times - 1 - 2 \times - 1) + 6 (5 \times - 1 - 2 \times - 1)$

$d e t (B) = 2 (- 3) - 4 (- 5) + 6 (- 3)$

$d e t (B) = - 6 + 20 - 18 = 0$

Problem 3: Find the determinant of the matrix:

C = $321132243$

Solution: Here, we will perform a column operation to simplify the matrix:

Subtract 2 multiplied by the first column from the second column:

$C_{2} \to C_{2} - 2 C_{1}$

$321 - 5 - 3 0 243$

Calculate the simplified matrix along the first row

$d e t (C) = 3 \times - 3 0 43 - (- 5) \times 2143 + 2 \times 21 - 3 0$

$d e t (C) = 3 (- 9 - 0) + 5 (6 - 4) + 2 (0 + 3)$

$d e t (C) = - 27 + 10 + 6 = - 27 + 16 = - 11$

Frequently Asked Questions

Does the determinant of a triangular matrix depend on size?

Can the determinant be used to solve systems of equations?

What happens if two rows or columns of a matrix are identical?

How do we compute a larger determinant?

Frequently Asked Questions

Does the determinant of a triangular matrix depend on size?

Can the determinant be used to solve systems of equations?

What happens if two rows or columns of a matrix are identical?

How do we compute a larger determinant?

Join ALLEN!

Properties of Determinants

1.0Determinants of a Square Matrix

2.0Properties of Determinants

3.0Solved Examples

Table of Contents

Join ALLEN!

Properties of Determinants

1.0Determinants of a Square Matrix

2.0Properties of Determinants

3.0Solved Examples

Table of Contents