Exploration: Consider the matrix A=( 2 7 1 -1 3 2 5 5 -4

Exploration: Consider the matrix A=( 2 7 1 ...

Exploration: Consider the matrix $A=\left(\begin{array}{rrr}2 & 7 & 1 \\ -1 & 3 & 2 \\ 5 & 5 & -4\end{array}\right)$ Define $f(x)=\operatorname{det}(x I-A)$ where $x$ is any real number and $I$ is the identity matrix. a) Find $\operatorname{det}(A)$ b) Expand $f(x)$ and compare the constant term to your answer in a). c) How is the coefficient of $x^{2}$ in the expansion of $f(x)$ related to $A ?$

Exploration:
Consider the matrix $A=\left(\begin{array}{rrr}2 & 7 & 1 \\ -1 & 3 & 2 \\ 5 & 5 & -4\end{array}\right)$ Define $f(x)=\operatorname{det}(x I-A)$ where $x$ is
any real number and $I$ is the identity matrix.
a) Find $\operatorname{det}(A)$
b) Expand $f(x)$ and compare the constant term to your answer in a).
c) How is the coefficient of $x^{2}$ in the expansion of $f(x)$ related to $A ?$

Key Concepts

Trace of a Matrix

The trace of a matrix is the sum of its diagonal elements and appears prominently in the coefficients of the characteristic polynomial. For a 3×3 matrix, the coefficient of x² in the characteristic polynomial is the negative of the trace of A. This relationship connects the algebraic properties of the matrix to its eigenvalues and provides insight into the behavior of the matrix under various operations.

Determinant

The determinant of a square matrix is a scalar value that summarizes certain properties of the matrix, such as invertibility. It is computed from the elements of the matrix and remains unchanged under matrix similarity transformations. In this context, the determinant of A is directly linked to the constant term of the characteristic polynomial, reflecting the product of the eigenvalues of A.

Characteristic Polynomial

The characteristic polynomial of a matrix, defined as f(x) = det(xI - A), is a polynomial whose roots are the eigenvalues of the matrix. This polynomial encapsulates several intrinsic properties of the matrix, including its determinant and trace. The constant term of this polynomial, when expanded, is closely related to the determinant of A, and its coefficients are connected to the sums and products of the eigenvalues.

Transcript

00:01 Hello students we are given a matrix we just need to find those abc part okay we will go with the first part we are asked for the determinant of the a okay so basically we can write two times of three three times of minus four comes out as minus 12 minus five to ja ten okay after that minus seven times of what we can do here minus one times of minus four comes out as four minus five two jar ten after that plus one time so here minus one times of five comes out as minus five okay minus five three ja fifteen now we just need to make a little bit calculation as well.

00:32 It is nothing about a determinant of a.

00:34 Now, determinant of a can be written as minus 12, minus 10 comes out as minus 22 and it's twice is nothing about minus 44.

00:42 After the 4 minus 10 comes out as minus 6, so minus minus plus here 7 6 are 42.

00:49 Okay.

00:50 Now minus 5 minus 15 comes out as minus 20.

00:53 So basically it can be written as like that.

00:55 So determinant of a can be written as see minus 44 plus 40.

01:00 Comes out as minus 2.

01:02 And minus 2, minus 20 comes out as minus 22.

01:06 Now, we will go for the b part.

01:09 What we are given? the expand fx.

01:11 We just need to expand fx.