Statement 1 A=[ 4 -1 -4 3 0 -4 3 -1 -3 ] is an involutar

Statement 1 A=[ 4 -1 -4 3 0 -4 ...

Statement 1 $A=\left[\begin{array}{ccc}4 & -1 & -4 \\ 3 & 0 & -4 \\ 3 & -1 & -3\end{array}\right]$ is an involutary matrix. and Statement 2 A is an involutary matrix if $(I+A)(I-A)=0$ If $\mathrm{A}$ is a square matrix of order $\mathrm{n}$ and I represents the unit matrix of order $\mathrm{n},|\mathrm{A}-\lambda \mathrm{I}|=0$ (where $|\mathrm{A}-\lambda \mathrm{I}|$ is the determinant of the matrix $\mathrm{A}-\lambda \mathrm{I}$ ) is called the characteristic equation of $\mathrm{A}$. As an illustration, consider the second order matrix $\mathrm{A}=\left(\begin{array}{ll}3 & 3 \\ 1 & 5\end{array}\right)$. $$ A-\lambda I=\left(\begin{array}{ll} 3 & 3 \\ 1 & 5 \end{array}\right)-\lambda\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)=\left(\begin{array}{cc} 3-\lambda & 3 \\ 1 & 5-\lambda \end{array}\right) $$ Characteristic equation of $\mathrm{A}$ is given by $|\mathrm{A}-\lambda \mathrm{I}|=0$ $\Rightarrow(3-\lambda)(5-\lambda)-3=0$ $\Rightarrow \quad \lambda^{2}-8 \lambda+12=0$ The roots of the characteristic equation are called the characteristic values (or eigen values) of the matrix $\mathrm{A}$. For the above matrix A, characteristic values are 2 and 6 . Results (i) Sum of the characteristic values of $\mathrm{A}=$ sum of the diagonal elements of $\mathrm{A}$, (the 'trace' of $\mathrm{A}$ ). (ii) Product of the characteristic values of $\mathrm{A}=|\mathrm{A}|$.

Statement 1
$A=\left[\begin{array}{ccc}4 & -1 & -4 \\ 3 & 0 & -4 \\ 3 & -1 & -3\end{array}\right]$ is an involutary matrix.
and
Statement 2 A is an involutary matrix if $(I+A)(I-A)=0$
If $\mathrm{A}$ is a square matrix of order $\mathrm{n}$ and I represents the unit matrix of order $\mathrm{n},|\mathrm{A}-\lambda \mathrm{I}|=0$ (where $|\mathrm{A}-\lambda \mathrm{I}|$ is the determinant of the matrix $\mathrm{A}-\lambda \mathrm{I}$ ) is called the characteristic equation of $\mathrm{A}$. As an illustration, consider the second order matrix $\mathrm{A}=\left(\begin{array}{ll}3 & 3 \\ 1 & 5\end{array}\right)$.
$$
A-\lambda I=\left(\begin{array}{ll}
3 & 3 \\
1 & 5
\end{array}\right)-\lambda\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right)=\left(\begin{array}{cc}
3-\lambda & 3 \\
1 & 5-\lambda
\end{array}\right)
$$
Characteristic equation of $\mathrm{A}$ is given by $|\mathrm{A}-\lambda \mathrm{I}|=0$
$\Rightarrow(3-\lambda)(5-\lambda)-3=0$
$\Rightarrow \quad \lambda^{2}-8 \lambda+12=0$
The roots of the characteristic equation are called the characteristic values (or eigen values) of the matrix $\mathrm{A}$. For the above matrix A, characteristic values are 2 and 6 .
Results
(i) Sum of the characteristic values of $\mathrm{A}=$ sum of the diagonal elements of $\mathrm{A}$, (the 'trace' of $\mathrm{A}$ ).
(ii) Product of the characteristic values of $\mathrm{A}=|\mathrm{A}|$.

Key Concepts

Trace

The trace of a matrix is the sum of its diagonal elements and possesses the key property of being equal to the sum of the eigenvalues of the matrix. It is a simple yet powerful tool in matrix analysis that remains invariant under similarity transformations and helps in various proofs and applications within linear algebra.

Determinant

The determinant is a scalar attribute of a square matrix that summarizes several aspects of the matrix, including whether it is invertible or singular. In the context of eigenvalues, the determinant of the matrix equals the product of its eigenvalues, serving as an important invariant in matrix theory.

Characteristic Equation

The characteristic equation of a matrix is formed by setting the determinant of (A - ?I) equal to zero, where A is a square matrix, ? is a scalar, and I is the identity matrix of the same order. This equation is fundamental in linear algebra as it is used to determine the eigenvalues of a matrix, thereby revealing essential properties of the corresponding linear transformation.

Involutory Matrix

An involutory matrix is a matrix that is its own inverse, meaning that when it is squared, the result is the identity matrix. This property, expressed as A² = I, implies unique structural features in the matrix, particularly concerning its eigenvalues which are typically ±1, and has applications in reflections and symmetry operations in linear algebra.

Eigenvalues

Eigenvalues, also known as characteristic values, are the scalars obtained as the solutions of the characteristic equation. They provide deep insights into the behavior of linear transformations, such as stability and diagonalizability, and are central to many applications in differential equations, physics, and engineering.

Transcript

00:01 So in the given question what we have is two statements and in the first statement what we are told is that we have a matrix a which is given as 4 minus 1 minus 4, 3 minus 4, 3 minus 1 minus 3 and we are told that this matrix is an involuntary matrix.

00:32 This matrix is an involuntary matrix.

00:39 And we have another statement in this question, which says that the matrix a is an involuntary matrix.

00:51 It is an involuntary matrix.

00:56 If it satisfies the condition, i plus a times i minus a, is equal to 0.

01:12 So these are the two statements that are given in the question.

01:16 So what we can do here is we can see that statement 2 is in fact the property or we can take it as a checking a verification of a matrix if it is involuntary or not right is it is a it is a property of an involuntary matrix that if we take the property the primary matrix that if we take the product of i plus a and i minus a we would get the zero matrix.

01:46 So this is in fact a property of a matrix, right? so now what we can say that we can say that the statement two is a true statement.

01:59 So in the question what we are told to do is to check whether statement two and one are true or false and if they have any relation between two between the two statements statements we should state that also right so we have just mentioned that since 2 is a standard property of an involuntary matrix 2 is a true statement and to check whether whether one the statement 1 is true we can just first expand the statement right so if i plus a times i minus a is equal to zero what we actually have over here is i squared minus a square equal to zero or we can write a square is equal to i right so this is the property by which we can write what an involuntary matrix is so if a is an involuntary matrix when we take the square of a we would get the identity matrix...

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