The number of singular matrices of order 2, whose elements are from the set {2,3,6,9}, is _____
Get the complete, step-by-step math solution for: "The number of singular matrices of order 2, whose elements are from the set \{2, 3, 6, 9\}, is _____". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Define a singular matrix
A square matrix A is singular if its determinant is equal to zero. For a 2×2 matrix, the determinant is calculated as ad - bc. So, we are looking for matrices where ad−bc=0, which means ad=bc. The elements a, b, c, d must come from the set {2,3,6,9}.
Step 2: List possible products
We need to find pairs of elements (a, d) and (b, c) from the set {2,3,6,9} such that their products are equal. Let's list all possible products of two elements from this set. The set of elements is S={2,3,6,9}.
Step 3: Calculate all possible products
Let's create a multiplication table for the elements in the set {2,3,6,9} to find all possible products. This table shows all 4×4=16 possible products when choosing two elements from the set, with replacement and order matters for the elements in the product.
Step 4: Identify equal products and count matrices
Now we need to find pairs of products (ad, bc) that are equal. We will count the number of ways to choose a, d, b, c such that ad=bc. We can categorize these cases based on the value of the product:
1. Product = 4: Only 2×2=4. So ad=4 has 1 way ((a,d)=(2,2)) and bc=4 has 1 way ((b,c)=(2,2)). This gives 1×1=1 matrix.
2. Product = 6: 2×3=6 and 3×2=6. So ad=6 has 2 ways, and bc=6 has 2 ways. This gives 2×2=4 matrices.
3. Product = 9: 3×3=9. So ad=9 has 1 way, and bc=9 has 1 way. This gives 1×1=1 matrix.
4. Product = 12: 2×6=12 and 6×2=12. So ad=12 has 2 ways, and bc=12 has 2 ways. This gives 2×2=4 matrices.
5. Product = 18: 2×9=18, 3×6=18, 6×3=18, 9×2=18. So ad=18 has 4 ways, and bc=18 has 4 ways. This gives 4×4=16 matrices.
6. Product = 27: 3×9=27 and 9×3=27. So ad=27 has 2 ways, and bc=27 has 2 ways. This gives 2×2=4 matrices.
7. Product = 36: 6×6=36. So ad=36 has 1 way, and bc=36 has 1 way. This gives 1×1=1 matrix.
8. Product = 54: 6×9=54 and 9×6=54. So ad=54 has 2 ways, and bc=54 has 2 ways. This gives 2×2=4 matrices.
9. Product = 81: 9×9=81. So ad=81 has 1 way, and bc=81 has 1 way. This gives 1×1=1 matrix.
Summing these up: 1+4+1+4+16+4+1+4+1=36.