Show that if A, B ∈ℂ^n ×n and A B=In, then for every λ∈ℂ b11+b21 λ+⋯+bn 1 λ^n-1=(det[ 1 λ

step 1 In this example, we have been given three matrices, A, B and C, and each of them is an N cross N

Show that if A, B ∈ℂ^n ×n and A B=In, then for every λ∈ℂ...

Show that if $A, B \in \mathbb{C}^{n \times n}$ and $A B=I_n$, then for every $\lambda \in \mathbb{C}$ $$ b_{11}+b_{21} \lambda+\cdots+b_{n 1} \lambda^{n-1}=\frac{\operatorname{det}\left[\begin{array}{cccc} 1 & \lambda & \cdots & \lambda^{n-1} \\ a_{21} & a_{22} & \cdots & a_{2 n} \\ \vdots & & & \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{2 n n} \end{array}\right]}{\operatorname{det} A} . $$

Show that if $A, B \in \mathbb{C}^{n \times n}$ and $A B=I_n$, then for every $\lambda \in \mathbb{C}$
$$
b_{11}+b_{21} \lambda+\cdots+b_{n 1} \lambda^{n-1}=\frac{\operatorname{det}\left[\begin{array}{cccc}
1 & \lambda & \cdots & \lambda^{n-1} \\
a_{21} & a_{22} & \cdots & a_{2 n} \\
\vdots & & & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{2 n n}
\end{array}\right]}{\operatorname{det} A} .
$$

Key Concepts

Multilinearity of the Determinant

The determinant function is multilinear, meaning it is linear in each row when the other rows are held fixed. This property allows for the manipulation of determinants when rows are expressed as linear combinations, and it is instrumental in understanding how changes in specific entries or substitutions of rows affect the overall determinant value.

Cofactor Expansion and the Adjugate Matrix

Cofactor expansion is a method for computing determinants by developing them along a row or column. The adjugate of a matrix, composed of these cofactors, forms the cornerstone of one formula for finding the inverse of a matrix (B = adj(A)/det(A)). This approach connects the entries of the inverse directly to the minors and cofactors of the original matrix.

Matrix Inversion

This concept refers to finding a matrix B that, when multiplied with a given square matrix A, results in the identity matrix. In other words, if AB = I, then B is the inverse of A. This property is fundamental in solving systems of linear equations, understanding linear mappings, and performing various matrix manipulations in higher algebra.

Determinant

The determinant is a scalar attribute of a square matrix that encapsulates essential information about the matrix, such as whether it is invertible (a non-zero determinant) and how it scales volumes upon transformation. Its multiplicative property (det(AB) = det(A)det(B)) and its behavior under row operations are crucial in deriving many identities and formulas in linear algebra.

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