Problem 7
Question
Consider the set \(A\) of all determinants of order 3 with entries 0 or 1 only. Let \(B\) be subset of \(A\) consisting of all determinants with value \(1 .\) Let \(C\) be the subset \(A\) of consisting of all determinants with value \(-1\). Then (A) \(C\) is empty. (B) \(B\) has as many elements as \(C\). (C) \(A=B \cup C\). (D) \(B\) has twice as many elements as \(C\).
Step-by-Step Solution
Verified Answer
(A) C is empty.
1Step 1: Understanding Determinant with 0s and 1s
A determinant of order 3 is a 3x3 matrix, and its determinant is calculated using the formula: \[ \det(A) = a(ei − fh) − b(di − fg) + c(dh − eg) \] where each element \(a, b, c, d, e, f, g, h, i\) can be either 0 or 1 because the entries of the matrix consist only of 0s and 1s.
2Step 2: Possible Values of Determinants
A matrix with entries 0 or 1 can result in a determinant value of 0, 1, or -1. The value of 0 arises from singular matrices (matrices that aren't invertible), whereas the values 1 and -1 occur when the matrix is non-singular.
3Step 3: Exploring Subset C with Value -1
If a determinant value is -1, it means the matrix is invertible but does not satisfy symmetry or skew-symmetry necessary to yield -1 with just 0 and 1 entries. Examine if any permutation of rows or columns in such matrices could yield -1 with these constraints.
4Step 4: Verification of C's Elements
Try different permutations of ones and zeros in a 3x3 matrix to see if any determinant can be -1. Given the properties of determinants with binary entries, constructing an invertible matrix with a determinant exactly -1 is not possible without involving integers other than 0 or 1.
5Step 5: Relating Sets B and C
Since subset \(C\) is empty due to the impossibility of forming a determinant -1 matrix using just 0 and 1 entries, it implies \(B\), the non-empty subset of matrices with determinant 1, cannot have equal or twice the elements compared to \(C\).
6Step 6: Conclusion from Analyzing Subsets and Union
As \(C\) is empty, \(B\) contains all invertible matrices with binary entries. Hence \(A\) can't be represented strictly by the union of \(B\) and \(C\) as suggested initially.
Key Concepts
3x3 matricesbinary matrixnon-singular matricesmatrix invertibility
3x3 matrices
Understanding the structure of a 3x3 matrix is fundamental in comprehending determinants. A 3x3 matrix consists of three rows and three columns, containing nine elements in total.
Each element in the matrix can be denoted by letters such as a, b, c, d, e, f, g, h, i, organized like this:
Each element in the matrix can be denoted by letters such as a, b, c, d, e, f, g, h, i, organized like this:
- First row: a, b, c
- Second row: d, e, f
- Third row: g, h, i
binary matrix
A binary matrix is a specific kind of matrix where each entry is either 0 or 1. This simplification makes calculations involving determinants intriguing yet challenging.
The restriction to binary entries implies that the determinant calculations are limited to specific outcomes, primarily focusing on integer values like 0, 1, and at times -1.
Binary matrices hold special significance in areas like computer science and combinatorics, representing on/off or true/false states. The calculative exploration of determinants within these matrices helps in pinpointing matrix properties and behaviors, such as symmetry and skew-symmetry.
Such constraints make determining values like -1 unlikely without expanding the integer possibilities beyond binary digits.
The restriction to binary entries implies that the determinant calculations are limited to specific outcomes, primarily focusing on integer values like 0, 1, and at times -1.
Binary matrices hold special significance in areas like computer science and combinatorics, representing on/off or true/false states. The calculative exploration of determinants within these matrices helps in pinpointing matrix properties and behaviors, such as symmetry and skew-symmetry.
Such constraints make determining values like -1 unlikely without expanding the integer possibilities beyond binary digits.
non-singular matrices
Non-singular matrices are those matrices with a non-zero determinant. These matrices are invertible, meaning they have an inverse matrix that, when multiplied with the original matrix, results in the identity matrix.
For a 3x3 binary matrix, achieving a determinant value of 1 is feasible, indicating that such a matrix is non-singular. The determinant fails to be zero, signifying that there are no linear redundancies among its rows or columns.
Non-singular matrices play a crucial role in linear algebra, as they allow for operations like matrix inversion, crucial for solving systems of linear equations and transforming spaces in geometry. By contrast, determining a binary matrix with a determinant of -1 proves impossible due to the restricted binary entries.
For a 3x3 binary matrix, achieving a determinant value of 1 is feasible, indicating that such a matrix is non-singular. The determinant fails to be zero, signifying that there are no linear redundancies among its rows or columns.
Non-singular matrices play a crucial role in linear algebra, as they allow for operations like matrix inversion, crucial for solving systems of linear equations and transforming spaces in geometry. By contrast, determining a binary matrix with a determinant of -1 proves impossible due to the restricted binary entries.
matrix invertibility
Matrix invertibility is determined by whether a matrix is non-singular. A matrix with a determinant of zero is singular and thus non-invertible.
On the other hand, non-singular matrices, particularly those with a determinant of 1 or -1 for non-binary matrices, are invertible. This allows for the creation of an inverse matrix.
In the context of binary 3x3 matrices, the focus is often on establishing the achievable determinant values, like 1, confirming invertibility. Having a determinant of 1 suggests an opportunity for reversibility within operations, critical in mathematical computations involving transformations.
A dependable understanding of invertibility aids students in appreciating underlying properties and limitations associated with matrices in linear algebra and its applications.
On the other hand, non-singular matrices, particularly those with a determinant of 1 or -1 for non-binary matrices, are invertible. This allows for the creation of an inverse matrix.
In the context of binary 3x3 matrices, the focus is often on establishing the achievable determinant values, like 1, confirming invertibility. Having a determinant of 1 suggests an opportunity for reversibility within operations, critical in mathematical computations involving transformations.
A dependable understanding of invertibility aids students in appreciating underlying properties and limitations associated with matrices in linear algebra and its applications.
Other exercises in this chapter
Problem 5
Let \(R\) be a relation defined on the set of natural numbers \(N\) as \(R=[(x, y): x \in N, y \in N, 2 x+y=41]\). Then (A) Domain of \(R=\\{1,2,3, \ldots, 19,2
View solution Problem 6
Let \(A=\\{x: x \in R,|x|
View solution Problem 8
Let \(A\) and \(B\) be two sets then \((A \cup B)^{\prime} \cup\left(A^{\prime} \cap B\right)\) is equal to (A) \(B^{\prime}\) (B) \(B\) (C) \(A\) (D) \(A^{\pri
View solution Problem 9
If \(A\) is the set of even natural numbers less than 8 and \(\mathrm{B}\) is the set of prime numbers less then 7 , then the number of relations from \(A\) to
View solution