Multiplication of Two Matrices

If\[A = \begin{bmatrix}\cos \alpha & \sin \alpha \\ - \sin \alpha & \cos \alpha\end{bmatrix}\] , then verify that A^T A = I₂.

Three schools X, Y, and Z organized a fete (mela) for collecting funds for flood victims in which they sold hand-helds fans, mats and toys made from recycled material, the sale price of each being Rs. 25, Rs. 100 andRs. 50 respectively. The following table shows the number of articles of each type sold:

Using matrices, find the funds collected by each school by selling the above articles and the total funds collected. Also write any one value generated by the above situation.

Given that A = `[(α,β),(γ,-α)]` and A² = 3I, then:

To raise money for an orphanage, students of three schools A, B and C organised an exhibition in their locality, where they sold paper bags, scrap-books and pastel sheets made by them using recycled paper, at the rate of Rs 20, Rs 15 and Rs 5 per unit respectively. School A sold 25 paper bags, 12 scrap-books and 34 pastel sheets. School B sold 22 paper bags, 15 scrap-books and 28 pastel sheets while School C sold 26 paper bags, 18 scrap-books and 36 pastel sheets. Using matrices, find the total amount raised by each school.

By such exhibition, which values are generated in the students?

For any two matrices A and B, we have ______.

If A and B are two matrices of the order 3 × m and 3 × n, respectively, and m = n, then the order of matrix (5A – 2B) is ______.

If   ` A =[a_ij]`    is a skew-symmetric matrix, then write the value of ` Σ_i  a_ij`

If \[A = \begin{bmatrix}n & 0 & 0 \\ 0 & n & 0 \\ 0 & 0 & n\end{bmatrix} \text {and B} = \begin{bmatrix}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c^1 & c_2 & c_3\end{bmatrix}\]then AB is equal to