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Problem 1. Compute the following determinant by anyway you like. (1) 140 122 1 4 2 (2) 1 22 2 44 1902 2345 ⇡ cos 2 1 (3) 2 5 1 2 4 1 2 1 2 1 2 1 4 (4) 1 1 1 1 1 1 1 1 1 (5) 1 2 8 9 (6) 9 0 3 9 (7) 2 3 6 1 0 0 0 0 0 0 0 0 1 2 5 7 (8) 0 0 0 0 3 6 1 3 1 2 5 6 0 0 0 0 (9) 0 0 2 5 3 6 0 0 0 0 1 2 9 5 0 0 (Hint: try to switch some row?) 4 (10) 2 3 6 1 0 0 0 0 5 6 7 9 1 2 5 7 Problem 2. (1) Suppose we know det(A) = 2, what is det(A1)? Why is that? (2) Suppose we know det(I A) = 3, det(3I A) = 5, what is det(A2 4A + 3I)? (Hint: Expand (3I A)(I A)) 0 m i 1 Proble 3. Find the verse by method of computing the adjugate of the following matrices n (1) @ 1 3 5 1 0 2 4 1 2 A (2) ✓ a b c d (3) 0 @ 1 1 1 1 1 A Problem 4. Find the inverse by method of elementary row transformation of the following matrices (1) 0 @ 1 21 9 01 5 1 2 1 A (2) ✓ 2 3 5 9 (3) 0 BB@ 1 2 5 1 1 2 0 9 1 2 1 7 0 0 2 3 1 CCA Problem 5. In this exercise, we prove a fun fact that det(I AB) = det(I BA) Suppose A is m ⇥ n matrix, B is n ⇥ m matrix. by using the block matrix row and column transformation. (1) Find a block matrix P, such that P ✓ Im Am⇥n Bn⇥m In = ✓ Im AB B In (2) Find a block matrix Q, such that ✓ Im Am⇥n Bn⇥m In Q = ✓ Im Bn⇥m In BA (3) Show that det(Im AB) = det(In BA) (4) Calculate the determinant of the matrix: 0 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 0 1 1 8 (Hint: what is 0 BBBB@ 1 1 1 1 1 1 CCCCA 11111 ? ) Problem 6. In this exercise, we talk about the construction of complex numbers. (1) Suppose a is a real number, show that a2 0 (Discuss case by case: a 0, a 0) (2) Althogh an element of square -1 can not happen in real numbers, but it could happen in 2⇥2 matrices over R, Now if we denote J = ✓ 1 1 , check that J2 = I2 (3) The complex number ✓ a + bi means the 2 ⇥ 2 matrix over R, which is a + bi = aI + bJ = a b b a . Proof the rule (a + bi)(c + di)=(ac bd)+(ad + bc)i 9 p (4) The absolute value of a complex number a + bi is defined to be |a + bi| = det(aI + bJ), show that this definition is coincide with the absolute value for real numbers. And it saitisfies |a + bi||c + di| = |(a + bi)(c + di)| (5) Show that if the absolute value of complex number is not equal to 0, then a + bi has reciprocal, the reciprocal is a a + bi bi | |2

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