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Difficulty: Very hardMatrices and Determinants

Let A=(21−13)A = \begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix} and B=(x21y)B = \begin{pmatrix} x & 2 \\ 1 & y \end{pmatrix} be 2×22 \times 2 matrices with integer entries xx and yy. If the matrix product ABAB is singular and det⁡(A+B)=0\det(A + B) = 0, what is the value of x+yx + y?

  1. −3-3Answer
  2. B
    33
  3. C
    −5-5
  4. D
    77

Answer

The value of x+yx + y is −3-3.
Since matrix ABAB is singular, det⁡(AB)=det⁡(A)det⁡(B)=0\det(AB) = \det(A)\det(B) = 0. Evaluating det⁡(A)=7≠0\det(A) = 7 \neq 0, we find det⁡(B)=xy−2=0\det(B) = xy - 2 = 0, giving xy=2xy = 2. Computing A+B=(2+x303+y)A+B = \begin{pmatrix} 2+x & 3 \\ 0 & 3+y \end{pmatrix}, its determinant is (2+x)(3+y)=6+3x+2y+xy=0(2+x)(3+y) = 6 + 3x + 2y + xy = 0. Substituting xy=2xy = 2 yields 3x+2y=−83x + 2y = -8. The integer pair satisfying both xy=2xy = 2 and 3x+2y=−83x + 2y = -8 is x=−2x = -2 and y=−1y = -1. Therefore, x+y=−3x + y = -3.

Step-by-Step Solution

1
Calculate the determinant of matrix AA.
det⁡(A)=(2)(3)−(1)(−1)=6+1=7\det(A) = (2)(3) - (1)(-1) = 6 + 1 = 7.
Knowing det⁡(A)\det(A) helps simplify the condition det⁡(AB)=0\det(AB) = 0 using determinant properties.
2
Use the singularity of ABAB to find a relationship between xx and yy.
det⁡(AB)=det⁡(A)det⁡(B)=7(xy−2)=0  ⟹  xy=2\det(AB) = \det(A)\det(B) = 7(xy - 2) = 0 \implies xy = 2.
The determinant of a product of matrices equals the product of their determinants.
3
Form the matrix A+BA + B and compute its determinant.
A+B=(2+x303+y)  ⟹  det⁡(A+B)=(2+x)(3+y)−(3)(0)=(2+x)(3+y)=6+3x+2y+xyA + B = \begin{pmatrix} 2+x & 3 \\ 0 & 3+y \end{pmatrix} \implies \det(A+B) = (2+x)(3+y) - (3)(0) = (2+x)(3+y) = 6 + 3x + 2y + xy.
Setting this determinant to zero gives a second equation involving xx and yy.
4
Substitute xy=2xy = 2 into det⁡(A+B)=0\det(A+B) = 0 and solve for integer values of xx and yy.
6+3x+2y+2=0  ⟹  3x+2y=−86 + 3x + 2y + 2 = 0 \implies 3x + 2y = -8. Testing integer pairs (x,y)(x,y) for xy=2xy = 2 gives x=−2x = -2 and y=−1y = -1.
3(−2)+2(−1)=−83(-2) + 2(-1) = -8 is satisfied only by x=−2,y=−1x = -2, y = -1 among integer pairs.
5
Compute x+yx + y.
x+y=−2+(−1)=−3x + y = -2 + (-1) = -3.
This answers the question directly.

Key Concept

Determinants of matrix products and sums, matrix singularity, and integer solutions to matrix equations.
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Topics

Matrices and Determinants

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