Construct a matrix whose column space contains (1, 1, 5) and (0, 3, 1) while it’s null space contains (1, 1, 2).

This question aims to understand the construction of a matrix under given constraints. To solve this question, we need to have a clear understanding of the terms column space and null space. The space which is spanned by the column vectors of a given matrix is called its column space. The space which is spanned […]

Describe all solutions of Ax=0 in parametric vector form, where A is row equivalent to the given matrix.

This problem aims to get us familiar with vector solutions. To better understand this problem, you should know about the homogeneous equations, parametric forms, and the span of vectors. We can define parametric form such that in a homogeneous equation there are $m$ free variables, then the solution set can be represented as the span of […]

For the matrix A below, find a nonzero vector in nul A and a nonzero vector in col A.

[ A = begin{bmatrix} 1 & -2 & 5 & 6 \ 5 & 1 & -10 & 15 \ 1 & -2 & 8 & 4 end{bmatrix} ] This question aims to find the null space which represents the set of all solutions to the homogeneous equation and column space which represents the range of a […]

Find x such that the matrix is equal to its own inverse.

[ M=left[ begin{matrix}7&x\-8&-7\end{matrix} right]] The aim of the article is to find the value of the variable $x$ within the given matrix for which it will be equal to its inverse matrix. The basic concept behind this question is the understanding of the Matrix, how to find the determinant of a matrix, and the inverse […]

A and B are n x n matrices. Mark each statement True or False. Justify your answer.

A row replacement operation does not affect the determinant of a matrix. The determinant of $A$ is the product of the pivots in any echelon form $U$ of $A$, multiplied by $(-1)^r$, where $r$ is the number of row interchanges made during row reduction from $A$ to $U$. If the columns of $A$ are linearly […]

Determine the value of h such that the matrix is the augmented matrix of a consistent linear system.

[ boldsymbol{ left[ begin{array}{ c c | c } 1 & 3 & -8 \ -4 & h & 1 end{array} right] } ] The aim of this question is to understand the solution of the system of linear equations using the row operations and row echelon form. Any matrix is said to be in […]

Find a basis for the space of 2×2 lower triangular matrices.

The main objective of this question is to find the basis space for the lower triangular matrices. This question uses the concept of basis space. A set of vectors B is referred to as a basis for a vector space V if each element of V can be expressed as a linear combination of finite […]

find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at (1, 3, 0), (-2, 0, 2),(-1, 3, -1).

This problem aims to find the volume of a parallelepiped, whose one vertex is at the origin (0,0) and the other 3 vertices are given. To solve this problem, it is required to have knowledge of 3-dimensional shapes along with their areas and volumes and to calculate determinants of the 3×3 square matrix. Expert Answer A […]

Determine if the columns of the matrix form a linearly independent set. Justify each answer.

(begin{bmatrix}1&4&-3&0\-2&-7&4&1\-4&-5&7&5end{bmatrix}) The main objective of this question is to determine whether the columns of the given matrix form a linearly independent or dependent set. If the non-trivial linear combination of vectors equals zero, then the set of vectors is said to be linearly dependent. The vectors are said to be linearly independent if there is […]