If A and B are nonsingular matrices of the same order, show that adj(A B)=(adj B)(adj A) . Verif

If A and B are nonsingular matrices of the same order, show...

Key Concepts

Properties of Matrix Multiplication

Matrix multiplication is associative and, when combined with the properties of determinants and adjugate matrices, helps in establishing relationships such as adj(AB) = (adj B)(adj A). Understanding the order of multiplication and how each operation affects the resulting matrix is vital in proving and verifying these types of algebraic identities.

Matrix Singularity and Nonsingularity

A matrix is nonsingular if it has a nonzero determinant and an inverse; it is singular if its determinant is zero, meaning no inverse exists. This distinction is important because many identities, including those involving the adjugate, have straightforward proofs under nonsingularity, while the singular case often requires additional verification or consideration.

Determinant

The determinant of a matrix is a scalar value that provides important information about the matrix, such as whether it is invertible (nonsingular) or not (singular). It satisfies the multiplicative property, where det(AB) equals det(A) multiplied by det(B), which is essential in the derivation of identities involving adjugate matrices.

Adjugate Matrix

The adjugate (or classical adjoint) of a matrix is defined as the transpose of the cofactor matrix. It plays a critical role in expressing the inverse of a matrix, especially in the formula A?¹ = (1/det(A)) adj(A) for nonsingular matrices. This concept is central when dealing with matrix identities like adj(AB) = (adj B)(adj A), as it underpins the relationship between matrix products and their cofactors.

Matrix Inverse

For a nonsingular matrix, the inverse is defined via the equation A?¹ = (1/det(A)) adj(A). This connection between the inverse and the adjugate matrix is crucial for proving relations like adj(AB) = (adj B)(adj A) and understanding how matrix products can be decomposed in terms of their cofactors.

Transcript

00:01 Hi, so today we are going to prove adjoined of ab equal to adjoined of b into adjoined of a.

00:07 In order to prove this first we are writing the results which are familiar to us.

00:13 A, b, whole inverse equal to adjoined of a b divided by determinant of a b.

00:22 This is the result we know.

00:24 From this we can write adjoined of a .b equal to a .b whole inverse into determinate.

00:32 Of ab.

00:33 Let it be equation number 1.

00:38 Okay then we have a b whole inverse equal to b inverse a inverse substituting and we have determinant of ab equal to d -a into d to b substitute these two in equation number one then we have adjoined of a equal to b inverse a inverse into determinant of a a into determinant of b.

01:06 Equation number 2.

01:09 Now, we have a inverse equal to adjoined of a by determinant of a.

01:18 B inverse equal to adjoined of b divided by determinant of b.

01:25 From this, we get adjoined of a equal to a inverse into determinant of a, adjoint of a, adjoint of.

01:34 B equal to b inverse into determinant of b and from this adjoined of b into adjoined of a is equal to b inverse determinant of b into a inverse determinant of a rearranging the terms we can write b inverse a inverse into determinant a determinant b this is equation number 3.

02:05 From the rhs of equation 2 and 3, we get adjoined of ab as equal to adjoined of b into adjoined of a.

02:19 Hence the proof.

02:21 Now we can verify this using some example...

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