Determinants

Step 1: Understanding the diagonal matrix.
A diagonal matrix is a square matrix in which the elements outside the main diagonal are zero. The determinant of a diagonal matrix is the product of the elements on the main diagonal. Step 2: Applying the determinant formula.
For the diagonal matrix , the determinant is: Step 3: Conclusion.
Thus, the determinant of the matrix is the product of its diagonal elements, corresponding to option (c).

Step 1: Using the property of determinants.
For a square matrix of order , if is multiplied by a scalar , then: where is the order of the matrix. Step 2: Applying this to the given matrix.
Since is a 3x3 matrix, we have , and the scalar is 2. Thus: Step 3: Conclusion.
Thus, , corresponding to option (c).

Step 1: Formula for the area of a triangle using determinants.
The area of a triangle with vertices , , and is given by the determinant formula: Substitute the coordinates of the points , , and : Step 2: Calculating the determinant.
Now, calculate the determinant: Simplifying each 2x2 determinant: Substitute these into the determinant formula: Simplify: We are given that the area is 12 square units, so: Multiply both sides by 2: This gives two cases: Step 3: Solving for .
Solving : Solving : Thus, the possible values of are 5 and -19. Step 4: Conclusion.
The value of is either 5 or -19.