Mathematics 10 — Exercise 3.2 · Q6 (i)

Full question

Verify (Aᵗ)ᵗ = A.

Step-by-step solution

1. Compute the left side first, following the bracket order.

2. Transpose: row 1 becomes column 1, row 2 becomes column 2, and so on.

   $$\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$$

3. Move row 1 into column 1.

   $$\left[\begin{array}{c}2\\ 3\end{array}\right]$$

4. Move row 2 into column 2.

   $$\left[\begin{array}{c}4\\ 5\end{array}\right]$$

5. Place the calculated entries in their original row and column positions.

   $$\left[\begin{array}{cc}2& 4\\ 3& 5\end{array}\right]$$

6. Transpose: row 1 becomes column 1, row 2 becomes column 2, and so on.

   $$\left[\begin{array}{cc}2& 4\\ 3& 5\end{array}\right]$$

7. Move row 1 into column 1.

   $$\left[\begin{array}{c}2\\ 4\end{array}\right]$$

8. Move row 2 into column 2.

   $$\left[\begin{array}{c}3\\ 5\end{array}\right]$$

9. Place the calculated entries in their original row and column positions.

   $$\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$$

10. Now compute the right side independently.

11. Every corresponding entry is equal, so the stated identity holds.

    $$(\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right],\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right])$$

Answer

Every corresponding entry is equal, so the stated identity holds.