Matrix Algebra Quiz

Matrix Determinant

• The determinant of a 3x3 matrix is calculated using a specific formula for the elements of the matrix.
• The problem asks to find the determinant of matrix A given as: A = $\begin{bmatrix} 1 & 0 & 2k \ -√k & -2√k & 0 \ 0 & 0 & √k \end{bmatrix}$

Coefficient of a Point

• The coefficient of a point with respect to a line is calculated using a specific formula.
• The problem asks to find the value of x given the coefficient of point (1, 2) is -4 with respect to it.

Matrix Multiplication and Addition

• The order of matrices influences the result of matrix multiplication and addition.
• The problem asks for the order of (A+B)TC, given the orders of A (3×4), B (4×3), and C (2×3)

Trace of a Matrix

• The trcae of a matrix is the sum of its diagonal elements
• The problem asks what the trace (A) is for matrix A = [aij], which is a 3x3 matrix.

Matrix Function

• A matrix function allows applying a function to each element of a matrix.
• Problem provides matrix A = $\begin{bmatrix} 2 & 1 & 5 \ -4 & -7 & 5 \end{bmatrix}$, and function f(t) = t2 - 5t + 3, asking to calculate f(A).

Matrix Inversion

• The inverse of a matrix can be calculated using various methods.
• Problem provides matrix A = $\begin{bmatrix} 1 & -3 \ -1 & 2 \end{bmatrix}$, and its inverse A^-1 = [3/5 1/5], asking to calculate A^-1.

System of Equations- Cramer's rule

• Cramer's rule provides a method to solve a system of linear equations by calculating determinants.
• The problem provides a system of three equations and asks to solve it using Cramer's rule.

Determinant Properties

• The determinant of a matrix can be proven without direct expansion using specific properties.
• The problem provides matrix A = $\begin{bmatrix} 1 & b & c\ a & 1 & b + c\ c & a+b & 1 \end{bmatrix}$, and asks to prove det A = 0 when a + b + c = 0.
• Further, the problem asks to show the determinant can be expressed as det(A) = (a²+ b²+c²)(a+b+c)(a-b)(b-c)(c-a).