Are Coefficient Matrices of the Systems of Linear Equations Nonsingular?
(a) Suppose that a $3\times 3$ system of linear equations is inconsistent. Is the coefficient matrix of the system nonsingular?
(b) Suppose that a $3\times 3$ homogeneous system of linear equations has a solution $x_1=0, x_2=-3, x_3=5$. Is the coefficient matrix of the system […]

Find the Vector Form Solution to the Matrix Equation $A\mathbf{x}=\mathbf{0}$
Find the vector form solution $\mathbf{x}$ of the equation $A\mathbf{x}=\mathbf{0}$, where $A=\begin{bmatrix}
1 & 1 & 1 & 1 &2 \\
1 & 2 & 4 & 0 & 5 \\
3 & 2 & 0 & 5 & 2 \\
\end{bmatrix}$. Also, find two linearly independent vectors $\mathbf{x}$ satisfying […]

Determine Trigonometric Functions with Given Conditions
(a) Find a function
\[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)\]
such that $g(0) = g(\pi/2) = g(\pi) = 0$, where $a, b, c$ are constants.
(b) Find real numbers $a, b, c$ such that the function
\[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 […]

If $M, P$ are Nonsingular, then Exists a Matrix $N$ such that $MN=P$
Suppose that $M, P$ are two $n \times n$ non-singular matrix. Prove that there is a matrix $N$ such that $MN = P$.
Proof.
As non-singularity and invertibility are equivalent, we know that $M$ has the inverse matrix $M^{-1}$.
Let us think backwards. Suppose that […]

An Example of a Real Matrix that Does Not Have Real Eigenvalues
Let
\[A=\begin{bmatrix}
a & b\\
-b& a
\end{bmatrix}\]
be a $2\times 2$ matrix, where $a, b$ are real numbers.
Suppose that $b\neq 0$.
Prove that the matrix $A$ does not have real eigenvalues.
Proof.
Let $\lambda$ be an arbitrary eigenvalue of […]

Two Matrices are Nonsingular if and only if the Product is Nonsingular
An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$.
Using the definition of a nonsingular matrix, prove the following statements.
(a) If $A$ and $B$ are $n\times […]