Consider the following in respect of matrices A and B of same order: 1) A 2– B 2= (A + B) (A – B) 2) (A – I) (I + A) = O ⇔ A 2= I Where I is the identity matrix and O is the null matrix.
2 only
Analyzing Matrix Properties: Statement 1
The first statement given is: \( A^2 - B^2 = (A + B) (A - B) \).
Let's expand the right-hand side of the equation, keeping in mind that matrix multiplication is not commutative in general (i.e., \( AB \neq BA \)).
\( (A + B) (A - B) = A(A - B) + B(A - B) \)
\( = A \cdot A - A \cdot B + B \cdot A - B \cdot B \)
\( = A^2 - AB + BA - B^2 \)
For the original statement \( A^2 - B^2 = (A + B) (A - B) \) to be true, we would need \( A^2 - B^2 = A^2 - AB + BA - B^2 \).
This simplifies to \( 0 = -AB + BA \), or \( AB = BA \).
The statement \( A^2 - B^2 = (A + B) (A - B) \) is only true if matrices A and B commute (i.e., \( AB = BA \)). Since the problem statement does not specify that A and B commute, this equation is not generally true for all matrices A and B of the same order.
Therefore, Statement 1 is incorrect in the general case.
Analyzing Matrix Properties: Statement 2
The second statement given is: \( (A - I) (I + A) = O \iff A^2 = I \), where \( I \) is the identity matrix and \( O \) is the null matrix.
Let's expand the left-hand side of the equivalence: \( (A - I) (I + A) \).
Using matrix multiplication rules:
\( (A - I) (I + A) = A(I + A) - I(I + A) \)
\( = A \cdot I + A \cdot A - I \cdot I - I \cdot A \)
Recall that for an identity matrix \( I \), \( A \cdot I = A \) and \( I \cdot A = A \), and \( I \cdot I = I \).
\( = A + A^2 - I - A \)
Simplifying the expression:
\( = A^2 + (A - A) - I \)
\( = A^2 + O - I \)
\( = A^2 - I \)
So, the equation \( (A - I) (I + A) = O \) is equivalent to \( A^2 - I = O \).
Adding \( I \) to both sides of the equation \( A^2 - I = O \), we get \( A^2 = I \).
Thus, \( (A - I) (I + A) = O \) is indeed equivalent to \( A^2 = I \).
Therefore, Statement 2 is correct.
Conclusion on Correct Statements
Based on our analysis:
- Statement 1: \( A^2 - B^2 = (A + B) (A - B) \) is generally incorrect for matrices as it requires \( AB = BA \).
- Statement 2: \( (A - I) (I + A) = O \iff A^2 = I \) is correct for matrices using properties of the identity matrix.
Thus, only Statement 2 is correct.
Revision Table: Matrix Properties Summary
| Property | General Statement | Matrix Equivalent | Condition for Matrices |
|---|---|---|---|
| Difference of Squares | \( a^2 - b^2 = (a-b)(a+b) \) | \( A^2 - B^2 \) | \( (A-B)(A+B) = A^2 + AB - BA - B^2 \) \( (A+B)(A-B) = A^2 - AB + BA - B^2 \) Equality holds iff \( AB = BA \) |
| Identity Expansion | \( (x-1)(x+1) = x^2-1 \) | \( (A-I)(A+I) \) or \( (A-I)(I+A) \) | \( (A-I)(I+A) = A^2 - I \) \( (A-I)(A+I) = A^2 + AI - IA - I^2 = A^2 + A - A - I = A^2 - I \) (Always true because \( AI=IA=A \) and \( I^2=I \)) |
Additional Information: Commutativity in Matrices
Matrix multiplication is generally not commutative. This means that for two matrices A and B, \( AB \) is usually not equal to \( BA \).
Commutativity (\( AB = BA \)) is a special property that holds only for certain pairs of matrices. For example, any matrix commutes with the identity matrix (\( AI = IA = A \)) and the null matrix (\( AO = OA = O \)). Diagonal matrices of the same order also commute with each other.
Properties from real number algebra, like \( a^2 - b^2 = (a-b)(a+b) \), rely on the commutative property of multiplication (\( ab = ba \)). When extending these properties to matrices, we must carefully check if the non-commutative nature of matrix multiplication affects the result.
In Statement 1, the expansion of \( (A+B)(A-B) \) results in terms involving both \( AB \) and \( BA \). If \( AB \neq BA \), these terms do not cancel out, and the expression does not simplify to \( A^2 - B^2 \).
In Statement 2, the expansion of \( (A-I)(I+A) \) involves terms like \( AI \) and \( IA \). Since \( AI = IA = A \) (identity matrix commutes with any matrix), these terms behave as expected, allowing the simplification to \( A^2 - I \).
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