(a) Find the determinant when a vector x replaces column j...

(a) Find the determinant when a vector $x$ replaces column $j$ of the identity (consider $x_{j}=0$ as a separate case): if $M=\left[\begin{array}{ccccc}1 & & x_{1} & & \\ & 1 & \vdots & & \\ & & x_{j} & & \\ & & \cdot & 1 & \\ & & x_{n} & & 1\end{array}\right]$ then $\operatorname{det} M=\longrightarrow$. (b) If $A x=b$, show that $A M$ is the matrix $B_{j}$ in equation ( 4 ), with $b$ in column $j$. (c) Derive Cramer's rule by taking determinants in $A M=B_{j}$.

(a) Find the determinant when a vector $x$ replaces column $j$ of the identity (consider $x_{j}=0$ as a separate case):
if $M=\left[\begin{array}{ccccc}1 & & x_{1} & & \\ & 1 & \vdots & & \\ & & x_{j} & & \\ & & \cdot & 1 & \\ & & x_{n} & & 1\end{array}\right]$ then $\operatorname{det} M=\longrightarrow$.
(b) If $A x=b$, show that $A M$ is the matrix $B_{j}$ in equation ( 4 ), with $b$ in column $j$.
(c) Derive Cramer's rule by taking determinants in $A M=B_{j}$.

Key Concepts

Determinant

The determinant is a scalar value computed from the elements of a square matrix, encapsulating key information about the matrix such as invertibility and volume scaling. Its computation relies on properties like multilinearity and alternating behavior with respect to the matrix's columns. These properties allow for manipulations such as replacing a column with another vector while still predicting the resulting determinant in terms of the original structure of the matrix.

Matrix Multiplication and Determinant Properties

Matrix multiplication connects the structure of different matrices, and one key property is that the determinant of a product of matrices equals the product of their determinants. This fact is used when a matrix with a replaced column is multiplied by another matrix, forming a new matrix whose determinant insightfully relates to that of the original, providing a bridge between operations on the identity matrix and their effects on the overall system.

Cramer’s Rule

Cramer’s Rule is a method for solving systems of linear equations using determinants. It involves replacing one column of the coefficient matrix with the constants vector from the system and then taking the ratio of the resulting determinant to the determinant of the original matrix. This rule leverages the multilinearity of the determinant with respect to its columns, offering an explicit formula for each variable in the system, thereby linking the abstract properties of determinants with practical solutions in linear algebra.

Column Replacement in Determinants

Column replacement involves substituting one of the columns of a given matrix with a different vector. Because the determinant is a multilinear function with respect to the columns, this operation can be analyzed by expressing the new determinant as a combination of the effects of the substitution. Recognizing how the determinant changes when one column is replaced is essential for understanding transformations of matrices used in solving linear systems.

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