### 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 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 […]

### Use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.

[ 1 + 2t^3, 2 + t – 3t^2, -t + 2t^2 – t^3] This problem aims to familiarize us with vector equations, linear independence of a vector, and echelon form. The concepts required to solve this problem are related to basic matrices, which include linear independence, augmented vectors, and row-reduced forms. To define linear […]

### Determine the head of the vector whose tail is given. Make a sketch.

– Given vector [ left[begin{matrix}-2\5\end{matrix}right] ] – Tail of the vector is $( -3, 2)$ [ left[begin{matrix}-3\2\end{matrix}right] ] In this question, we have to find the head of the vector when the vector and its tail are given. The basic concept behind this question is the knowledge of vectors, subtraction addition, and multiplication of […]

### Figure ABCD is a Trapezoid with point A (0, −4). What rule would rotate the figure 270° clockwise?

This question aims to find the type of rule that would be applied to the trapezoid ABCD with a point A( 0, -4 ) to rotate it to 270° in the clockwise direction. A quadrilateral having two sides parallel to each other is called a trapezoid. This four-sided figure is also called a trapezium. When […]