Question23 Suppose that [math] is a 2 by 2 matrix with two eigenvalues, [math] and [math]. Suppose that [math] is an eigenvector of [math] with eigenvalue [math] and that [math] is an eigenvector of [math] with eigenvalue [math]. Let [math]. Which of the following expressions will give [math]?Select one alternative: [math] [math] [math] [math] ResetMaximum marks: 1 Flag question undefined单项选择题
A
810[10]+610[11]
B
4⋅210[10]+2⋅310[11]
C
2⋅410[10]+3⋅210[11]
D
(210+310)(4[10]+2[11])

Question

Question23 Suppose that [math] is a 2 by 2 matrix with two eigenvalues, [math] and [math]. Suppose that [math] is an eigenvector of [math] with eigenvalue [math] and that [math] is an eigenvector of [math] with eigenvalue [math]. Let [math]. Which of the following expressions will give [math]?Select one alternative: [math] [math] [math] [math] ResetMaximum marks: 1 Flag question undefined单项选择题

A

810[10]+610[11]

B

4⋅210[10]+2⋅310[11]

C

2⋅410[10]+3⋅210[11]

D

(210+310)(4[10]+2[11])

BlackTom AI · Accepted Answer

Rewriting the problem in my own words: A is a 2x2 matrix with two eigenvalues, say λ1 and λ2, and corresponding eigenvectors v1 and v2. We are given that A v1 = λ1 v1 and A v2 = λ2 v2, and we are asked which expression gives A(2 v1 + 3 v2).

Option 1: 8·10[10] + 6·10[11]. If we interpret 10[10] and 10[11] as the coordinates of v1 and v2 in some basis, this option suggests a combination that scales v1 by 8 and v2 by 6, but the notation 10[10] and 10[11] is ambiguous and does not clearly reflect applying A to the linear combination 2v1 + 3v2. Moreover, the coefficients 8 and 6 align with applying A to 2v1 + 3v2 only if the eigenvalues are 4 and 2 respectively, but the way the components are written doesn’t explicitly show that A acts diagonally on v1 and v2 in the expression.

Option 2: 4·210[10] + 2·310[11]. This writes the result as 4·2·v1 + 2·3·v2 equals 8v1 + 6v2 in effect, if the intent is to multiply the coefficients by 2 and 3 before applying the eigenvalues. However, the placement of factors and the use of 210[10] and 310[11] is unconventional and could be misread as applying extra scaling to the basis vectors rather than applying A to the combination 2v1 + 3v2. It risks misunderstanding the order of operations or the intended representation of A acting on the eigenbasis.

Option 3: 2·4 10[10] + 3·2 10[11]. Here, interpret 10[10] as v1 and 10[11] as v2. This expression corresponds to A acting on the vector (2v1 + 3v2): A(2v1 + 3v2) = 2·(A v1) + 3·(A v2) = 2·(λ1 v1) + 3·(λ2 v2) = (2λ1) v1 + (3λ2) v2. If λ1 = 4 and λ2 = 2, this becomes (2·4) v1 + (3·2) v2 = 8 v1 + 6 v2, which matches the interpretation of the expression given and correctly applies the eigenvalues to the corresponding eigenvectors. The numerical factors line up exactly with the expected result of applying A to the linear combination 2v1 + 3v2.

Option 4: (210 + 310)(4[10] + 2[11]). This reads as a product of sums of basis components with a linear combination of eigenvector directions, which would mix the eigenvectors rather than apply A linearly to each eigencomponent separately. In other words, it doesn’t reflect A acting diagonally on the eigenbasis and thus does not correspond to A(2v1 + 3v2) in the straightforward way.

Putting it together, the correct approach to compute A(2v1 + 3v2) is to apply A to each eigencomponent using its eigenvalue, giving 2(λ1 v1) + 3(λ2 v2). With λ1 = 4 and λ2 = 2, that yields 8 v1 + 6 v2. Among the options, the expression that expands to 8 v1 + 6 v2 in the same notation is the one in Option 3: 2·4 10[10] + 3·2 10[11]. Therefore, Option 3 correctly represents A(2v1 + 3v2).