Matrices and Calculus - Fill in the Blanks

Fill-in-the-blank practice flashcards covering linear algebra, matrix operations, eigenvalues, quadratic forms, and mean value theorems from the lecture notes.

The rank of a unit matrix of order $n$ is __________.

$$n$$

If a square matrix is orthogonal, then $A^{-1} =$ __________.

$$A^T$$

The value of $K$ such that the given matrix has rank $2$ is __________.

$$4$$

The matrix $\begin{pmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{pmatrix}$ is __________.

an orthogonal matrix

The system of equations $AX = B$ is said to have a unique solution if __________.

$$|A| \neq 0$$

If $\lambda = -1, -2, 6$, then the signature of the matrix is __________.

$$(1, 2)$$

The diagonal matrix $D$ is __________.

$$\text{diag}(\lambda_1, \lambda_2, \dots, \lambda_n)$$

The eigenvalues of a symmetric matrix are __________.

real

The sum of the eigenvalues of a matrix is __________.

the trace of the matrix

The product of the eigenvalues of a matrix is __________.

the determinant of the matrix

Every square matrix satisfies its __________ equation.

Cayley-Hamilton

The eigenvalues of $A = \begin{pmatrix} a & h & g \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix}$ are __________.

$$a, b, c$$

A square matrix $A$ is said to be Hermitian if __________.

$$A = A^*$$

The nature of the canonical form $Y_1^2 + 2Y_2^2 - 3Y_3^2$ is __________.

indefinite

If $A = \text{diag}(2,1,3)$, the eigenvalues of $A^{-1}$ are __________.

$$\frac{1}{2}, 1, \frac{1}{3}$$

The rank of the given matrix is __________.

$$2$$

The nature of the quadratic form $2X^2 + 2Y^2 + 2Z^2 + 2YZ$ is __________.

positive definite

A homogeneous system of equations $AX = 0$ always has __________.

the trivial solution

If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, then $A^3 =$ __________.

$$\begin{pmatrix} 37 & 54 \\ 81 & 118 \end{pmatrix}$$

For $f(x) = x + \frac{1}{x}$ in $[\frac{1}{2}, 1]$, the value of $c$ from Rolle's theorem is __________.

Rolle's theorem is not applicable

Echelon form method involves only __________.

elementary row operations

The method used to compute $A^{-1}$ is called __________.

Gauss-Jordan method

The canonical form method is also known as __________.

normal form method

The eigenvector of a square matrix $A$ is a __________.

non-zero vector satisfying $AX = \lambda X$

State the Cayley-Hamilton theorem: Every square matrix satisfies its own __________.

characteristic equation

The sum of the eigenvalues of $A = \begin{pmatrix} 1 & 2 & 5 \\ 6 & 5 & 3 \\ 0 & 1 & 2 \end{pmatrix}$ is __________.

$$8$$

The diagonal matrix of a given square matrix $A$ is called __________.

diagonal form

The conclusion of Lagrange's Mean Value Theorem is __________.

$$f'(c) = \frac{f(b) - f(a)}{b - a}$$

The conclusion of Cauchy's Mean Value Theorem is __________.

$$\frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}$$

The rank of $\begin{pmatrix} 10 & 10 \\ 10 & 10 \end{pmatrix}$ is __________.

$$1$$

The rank of the identity matrix of order $6$ is __________.

$$6$$

The homogeneous linear system of equations in matrix form is __________.

$$AX = O$$

The inverse of $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is __________.

$$\frac{1}{2}\begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix}$$

The eigenvalues of $A = \begin{pmatrix} 5 & 0 \\ 0 & 7 \end{pmatrix}$ are __________.

$5$ and $7$

The characteristic equation of a square matrix $A$ is given by __________.

$$|A - \lambda I| = 0$$

The matrix of the real quadratic form $Q = 4x^2 + 6xy + 8y^2$ is __________.

$$\begin{pmatrix} 4 & 3 \\ 3 & 8 \end{pmatrix}$$

The condition for two square matrices $A$ and $B$ to be similar is __________.

$$A = P^{-1}BP$$

The conclusion of Rolle's theorem is __________.

$$f'(c) = 0$$

The domain/interval used to verify the mean value theorems is __________.

$$[a,b]$$