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Properties of the Determinant Function

MATH 322, Linear Algebra I

J. Robert Buchanan

Department of Mathematics

Spring 2007

Overview

Today’s discussion will illuminate some of the properties of the determinant: relationship with scalar products multi-linearity relationship with matrix products relationship with invertible matrices

Basic Properties

Theorem If A is an n × n matrix and k is a scalar, then det( kA ) = kn^ det( A ).

Proof.

A Non-property

We can show by example that det( A + B ) 6 = det( A ) + det( B ).

Example

Let A =

[

]

and B =

[

]

, then

det( A + B ) =

∣∣ = 4 6 = 1 + (− 2 ) = det( A ) + det( B ).

However...

If matrix A and B differ in just a single row...

Theorem Let A, B, and C be n × n matrices that differ only in a single row, say the r th^ row, and suppose the r th^ of C is the sum of the r th^ rows of A and B. Then det( C ) = det( A ) + det( B ).

Remark: the same result holds for columns.

Example

Let A =

[

a 11 a 12 a 21 a 22

]

and B =

[

a 11 a 12 b 21 b 22

]

, and

C =

[

a 11 a 12 a 21 + b 21 a 22 + b 22

]

, compare det( A ) + det( B ) to det( C ).

However...

If matrix A and B differ in just a single row...

Theorem Let A, B, and C be n × n matrices that differ only in a single row, say the r th^ row, and suppose the r th^ of C is the sum of the r th^ rows of A and B. Then det( C ) = det( A ) + det( B ).

Remark: the same result holds for columns.

Example

Let A =

[

a 11 a 12 a 21 a 22

]

and B =

[

a 11 a 12 b 21 b 22

]

, and

C =

[

a 11 a 12 a 21 + b 21 a 22 + b 22

]

, compare det( A ) + det( B ) to det( C ).

Determinant of Matrix Product

Theorem If A and B are n × n matrices then det( AB ) = det( A ) det( B ).

Remark: we will first prove this for the case where A is an elementary matrix.

Lemma Suppose E is an elementary n × n matrix and B is an n × n matrix, then det( EB ) = det( E ) det( B ).

Proof.

det( E 1 E 2 · · · EmB ) = det( E 1 ) det( E 2 ) · · · det( Em ) det( B )

Determinant of Matrix Product

Theorem If A and B are n × n matrices then det( AB ) = det( A ) det( B ).

Remark: we will first prove this for the case where A is an elementary matrix.

Lemma Suppose E is an elementary n × n matrix and B is an n × n matrix, then det( EB ) = det( E ) det( B ).

Proof.

det( E 1 E 2 · · · EmB ) = det( E 1 ) det( E 2 ) · · · det( Em ) det( B )

Determinant of Matrix Product

Theorem If A and B are n × n matrices then det( AB ) = det( A ) det( B ).

Remark: we will first prove this for the case where A is an elementary matrix.

Lemma Suppose E is an elementary n × n matrix and B is an n × n matrix, then det( EB ) = det( E ) det( B ).

Proof.

det( E 1 E 2 · · · EmB ) = det( E 1 ) det( E 2 ) · · · det( Em ) det( B )

Invertibility

Theorem A square matrix A is invertible if and only if det( A ) 6 = 0_._

Proof.

Remark: this is one of the most important results of linear algebra.

Invertibility

Theorem A square matrix A is invertible if and only if det( A ) 6 = 0_._

Proof.

Remark: this is one of the most important results of linear algebra.

Example

Example Determine if the matrix below is invertible.  

Return to Previous Theorem

We can now prove the general case of the theorem:

Theorem If A and B are n × n matrices then det( AB ) = det( A ) det( B ).

Proof. Case: A is singular. Case: A is invertible.

Return to Previous Theorem

We can now prove the general case of the theorem:

Theorem If A and B are n × n matrices then det( AB ) = det( A ) det( B ).

Proof. Case: A is singular. Case: A is invertible.