Consider the following statements about two square matrices A and B of the same order: P: (A+B)^

Consider the following statements about two square matrices...

Consider the following statements about two square matrices $\mathrm{A}$ and $\mathrm{B}$ of the same order: P: $(A+B)^{2}=A^{2}+2 A B+B^{2}$ Q: $(A+B)(A-B)=A^{2}-B^{2}$ Then, (a) both P and $\mathrm{Q}$ are true (b) both $\mathrm{P}$ and $\mathrm{Q}$ are false (c) both $\mathrm{P}$ and $\mathrm{Q}$ are true if $\mathrm{A}$ and $\mathrm{B}$ commute for multiplication (d) P is true but $\mathrm{Q}$ is false

Consider the following statements about two square matrices $\mathrm{A}$ and $\mathrm{B}$ of the same order:
P: $(A+B)^{2}=A^{2}+2 A B+B^{2}$
Q: $(A+B)(A-B)=A^{2}-B^{2}$
Then,
(a) both P and $\mathrm{Q}$ are true
(b) both $\mathrm{P}$ and $\mathrm{Q}$ are false
(c) both $\mathrm{P}$ and $\mathrm{Q}$ are true if $\mathrm{A}$ and $\mathrm{B}$ commute for multiplication
(d) P is true but $\mathrm{Q}$ is false

Key Concepts

Noncommutativity of Matrix Multiplication

In matrix algebra, the multiplication of matrices is generally not commutative, meaning that for two matrices A and B, AB does not necessarily equal BA. This fundamental property affects the validity of many algebraic identities that hold in the scalar case, as the order of factors is crucial in ensuring correct results.

Matrix Binomial Expansion

The binomial expansion for matrices, such as expanding (A+B)², requires careful handling because of the noncommutative nature of matrix multiplication. Unlike scalars, when expanding (A+B)², the cross terms cannot be combined unless the matrices commute, making the standard binomial expansion valid only under the assumption of commutativity.

Commutativity Condition for Matrix Identities

Many algebraic identities familiar from real numbers, such as (A+B)² = A² + 2AB + B² or (A+B)(A-B) = A² - B², hold true for matrices if and only if the matrices involved commute (that is, AB = BA). This condition is crucial for ensuring that terms can be rearranged and combined properly in the expansion or factorization.

Transcript

00:01 So in the given question we are told that there are two statements involving two squared matrices a and b.

00:11 So when a and b are given to be two squared matrices, right? so in the first statement we are told that when we take a plus b square it is equal to a square plus 2 ab plus 2a b plus b square where a and b are the matrices and in the statement two we are told that a plus b times a minus b is equal to a square minus b square so these are the statements that are given and we should check each of these statements to see whether each of whether these statements are true or false right so so what we have is we have a standard result of taking a plus b square in terms of a matrix, right? so when we are taking the sum of two matrices and the square of sum of two matrices, the formula is a plus b, all square is equal to a square plus ab plus b plus b squared plus b squared.

01:33 Right and it is only true for the statement p that is we can only say ab we can take 2 ab in this expression only if 2 ab only if ab and b are the same right and we can say p is true only under this condition and we can just say p is generally true right so since this condition is not mentioned in the statement p we would have to say that p is false right so next we have we have the statement q and in q we are told a plus b times a minus b is equal to a squared minus b square so in the same way a plus b times a minus b is also a standard result that is used in the form of matrices and when we are taking a plus b times a minus b four matrices the expansion is a squared plus b a minus ab plus b b square and we can take the statement q to be true only if only if a square minus b square only if b is equal to ab right so this can these are the conditions for p and q to be true right so this property that a b is equal to b is called the commutative property of multiplication, right? commutative property, commutative in multiplication.

03:46 So, when we say two matrices or two numbers are commutative in multiplication, what we mean is when we take a times b or b times a, we get the same answer.

03:59 So these two statements are true only in the condition that a and b are commutative in multiplication.

04:08 So there is an option in the question that says that both p and q are true are true if a and b commute for multiplication...

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