# Determinants Worksheet

Determinants are mathematical objects that have its application in analysis and solution of linear equations. It is a value that is determined from elements of a square matrix. Determinants are computed only for a square matrix. There exist a particular formula for 2×2 and 3×3 matrices.
In mathematics, a determinant is a scalar value that is calculated from the elements of the square matrix. The square matrix may be 2×2 matrix, 3×3 matrix, or nxn matrix. If “A” is a matrix, then the determinant of matrix A is given by det (A) or |A|.
Determinants Worksheets are very useful to find the determinants of the order three cross three (3*3) and two crosses two ( 2*2). Problems are based on CBSE Syllabus and NCERT guidelines. Students can solve the below problems on determinants. Solving the determinants problems help the students to score good marks in the final examination.

### Worksheet on Determinants

Solve the determinants problems given below:

 Compute the determinant for the following: $$\begin{vmatrix} 4 & -1\\ 6 & 2 \end{vmatrix}$$ $$\begin{vmatrix} 10 & 6\\ 4 & 5 \end{vmatrix}$$ Evaluate the determinant for the following: $$\begin{vmatrix} 6 & 4 & 2 \\ 3 & -7& 1\\ 5 & 5 & 3 \end{vmatrix}$$ $$\begin{vmatrix} 3 & 6 & 4 \\ 9 & 2 & 7\\ 6 & 5 & -1 \end{vmatrix}$$ Solve the system of equations using Cramer’s Rule: 2x+7y =-14, 9x+2y=55 3x-4y = 25, 3x+7y=14 Calculate the number of solutions for the given system of equations: 3x+4y = 1, 9x+12y = 3 7x-4y = 5, 6x+9y = 4 Solve the system of equations in three variables using Cramer’s Rule: (⅔)x+4y+(3/2)z = 34 (½)x-(3/2)y+2z=9 (3/2)x+(1/3)y-(½)z = -22 Prove that $$\begin{vmatrix} 1+a &1 &1 \\ 1 & 1+b & 1\\ 1 & 1 & 1+c \end{vmatrix} = abc+bc+ca+ab$$ Find the solution the given equation using Cramer’s Rule: (x/3)+(y/5)=6, (x/9)- (y/12)=-(¼) x = _____ y =_____ Find the value for the given determinant: $$\begin{vmatrix} 4 & a & b+c \\ 4 & b & c+a\\ 4 & c & a+b \end{vmatrix}$$ Show that $$\begin{vmatrix} 0 & a & -b \\ -a & 0 & -c \\ b & c & 0 \end{vmatrix} =0$$ using the properties of determinants. Show that $$\begin{bmatrix} b+c & a & a \\ b & c+a & b \\ c & c & a+b \end{bmatrix} = 4abc$$