The diagram below is made up of a right triangle and three squares.

Which of the following is represented by this diagram?

A) p2= r2-m2
B)p2=m2-r2
C)r2=p2-m2
D)r2=m2-p2

1





to see this. You should get a pivot in each column.
1
(c) No, they do not form a basis because they do not span P3. (You would need 4 vectors to span P3.)
3. Let M2×2 be the vector space of 2×2 matrices with the usual operations of addition and scalar multiplication.
Define the linear transformation
T : M2×2 → M2×2
T(A) = A + A
T
,
where AT
is the transpose of A.
(a) Find the matrix representation of T relative to the basis ��1 0
0 0�
,

0 1
0 0�
,

0 0
1 0�
,

0 0
0 1��.
(b) Find the dimension of the kernel of T.
Solution.
(a) The matrix representation is




2 0 0 0
0 1 1 0
0 1 1 0
0 0 0 2




.
(b) After row reducing the matrix from part (a) there is one free variable. So the dimension of the kernel is
1.
4.
(a) Compute the determinant
det




2 5 −3 −1
3 0 1 −3
−6 0 −4 9
4 10 −4 −1




.
(b) Let A, B be n × n matrices, such that B is invertible. Is it true that
det(B
−1AB) = det(A)?
Justify your answer!
Solution.
(a) The determinant is 120.
(b) It is true, since
det(B
−1AB) = det(B
−1
) det(A) det(B) = det(B
−1
) det(B) det(A) = det(I) det(A) = det(A).

To determine which option is represented by the given diagram, we need to understand the relationship between the sides of a right triangle and the squares that are constructed on those sides.

Let's denote the sides of the right triangle as follows:
- The side adjacent to the right angle as "r."
- The side opposite to the right angle as "m."
- The hypotenuse of the right triangle as "p."

According to the Pythagorean theorem, the relationship between the sides of a right triangle is given by the equation: p^2 = r^2 + m^2.

Looking at the diagram, we see that there are three squares constructed on each side of the right triangle.

Now, let's analyze the options:

A) p^2 = r^2 - m^2
This equation does not represent the relationship between the sides of a right triangle and the squares in the diagram. It is incorrect.

B) p^2 = m^2 - r^2
Again, this equation does not match the relationship between the sides of a right triangle and the squares shown in the diagram. It is incorrect.

C) r^2 = p^2 - m^2
This equation accurately represents the relationship between the sides of a right triangle and the squares in the diagram. The hypotenuse squared (p^2) is equal to the sum of the squares of the other two sides (r^2 + m^2). This option is correct.

D) r^2 = m^2 - p^2
This equation does not correctly represent the relationship between the sides of a right triangle and the squares in the diagram. It is incorrect.

Therefore, the correct answer is C) r^2 = p^2 - m^2.