Recent questions tagged system-of-equations

Which of the following represents the complete solution set of the given system?$$\begin{bmatrix}1 & 0 & 0 & 2 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & -1\end{bmatrix}\begin{bmatrix...

What values of $x, y$ and $z$ satisfy the following system of linear equations?$$\begin{bmatrix} 1 &2 &3 \\ 1& 3 &4 \\ 2& 2 &3 \end{bmatrix} \begin{bmatrix} x\\y \\ z \en...

For which vector(s) $b$ below does the system $ \begin{bmatrix}1 & 2 \\3 & 5 \\2 & 3\end{bmatrix} x = b $ have a solution?$ \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} $ $ ...

Let $Ax = b$ be a system of linear equations where $A$ is an $m \times n$ matrix and $b$ is a $m \times 1$ column vector and $X$ is an $n \times1$ column vector of unknow...

The given system has infinitely many solutions for $k=?$$$\left[\begin{array}{ccc}2 & 2 & -4 \\1 & 3 & -2 \\-4 & k & 8\end{array}\right]\left[\begin{array}{l}x \\y \\z\en...

Let $P_1$, $P_2$ and $P_3$ denote, respectively, the planes defined by$$\begin{array} {} a_1x +b_1y+c_1z=\alpha _1 \\ a_2x +b_2y+c_2z=\alpha _2 \\ a_3x +b_3y+c_3z=\alpha ...

The values of $a$ for which the following system of equations$$\begin{array} {} x & + & y & + & z & = & 1 \\ x & + & 2y & + & 4z & = & a \\ x & + & 4y & + & 10z & = & a ^...

Let two systems of linear equations be defined as follows:$\begin{array} {} & x+y & =1 \\ P: & 3x+3y & =3 \\ & 5x+5y & =5 \end{array}$ and $\begin{array} {} & x+y & =3 \\...

The $a, b, c$ and $d$ satisfy the equations$$\begin{matrix} a & + & 7b & + & 3c & + & 5d & = &16 \\ 8a & + & 4b & + & 6c & + & 2d & = &-16 \\ 2a & + & 6b & + & 4c & + & 8...

Consider a non-homogeneous system of linear equations representing mathematically an over determined system. Such a system will beconsistent having a unique solution cons...

Let $A$ be a $3 \times 5$ matrix $A$ and $\mathbf{b}$ in $\mathbb{R}^{3}$. Which of the following statements about the matrix equation $A \mathbf{x}=\mathbf{b}$ for $\mat...

Consider the system of equations$$\left\{\begin{array}{l}x+k y=1 \\k x+y=1\end{array}\right.$$For what value(s) of $k$ does this system become inconsistent, i.e., the two...

Consider the following system of linear equations:$x_1+2x_2-3x_3+4x_4=b_1,$$2x_1+5x_2-2x_3+x_4=b_2,$$5x_1+12x_2-7x_3+6x_4=b_3,$where \(b=\begin{bmatrix}b_1 & b_2 & b_3\en...

Consider the system of linear equations$$\begin{gathered}x+y+z=6 \\2 x+3 y+ z=11 \\x+2y+ 3z=11\end{gathered}$$The values of ( $x, y, z$ ) are$(1,2,3)$ $(2,1,3)$ $(3,1,2)$...

Let $A$ be an $m \times n$ matrix and $x$ and $b$ are column vectors.Which of the following is/are TRUE?If $A x=0$ has only the trivial solution $x=0$, then $A x=b$ alway...

Consider the system of linear equations given below.\[\begin{array}{c}a x+y=b \\16 x+a y=24\end{array}\]Suppose the values of $a$ and $b$ are chosen such that the system ...

Which one of the following is not a basic solution of system of linear equation$\begin{array}{l} x_{1}+2 x_{2}+x_{3}=4 \\ 2 x_{1}+x_{2}+5 x_{3}=5 \end{array}$$x_{1}=-1, x...

For what values of $x$:\[\left[\begin{array}{lll}1 & 2 & 1\end{array}\right]\left[\begin{array}{lll}1 & 2 & 0 \\2 & 0 & 1 \\1 & 0 & 2\end{array}\right]\left[\begin{array}...

Let $n \geq 3$ be an integer, and let $x_{1}, x_{2}, \ldots, x_{n}$ be variables which take real values with $0 \leq x_{i} \leq 1$ for all $1 \leq i \leq n$. Let$$\begin{...

A homogeneous linear system in n unknowns whose corresponding augmented matrix has a reduced row echelon form with r leading 1's hasn free variables$\mathrm{n}-\mathrm{r}...