Is it always true that AB = 0 means either A = 0 or B = 0? (2024)

Mathematics is not a contemplative but a creative subject; no one can draw much consolation from it when he has lost the power or the desire to create; and that is apt to happen to a mathematician rather soon. It is a pity, but in that case he does not matter a great deal anyhow, and it would be silly to bother about him.

G. H. Hardy

Is it always true that AB = 0 means either A = 0 or B = 0? (1)

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One of the important concepts in mathematics is the concept of zero and here we will consider an example involving zero. One well-known result is if the product of two (or more) quantities is equal to 0, then each of them is equal to zero.

Putting it in mathematical form means, if AB = 0 then either A = 0 or B = 0 or A = 0 = B. For example, if (x − 1)(x − 2) = 0 then we have, either (x − 1) = 0 or (x − 2) = 0, resulting in x = 1 or 2. Point to note, the mathematical word or is used here in the inclusive sense. Similar examples we may consider to explore it further. Is it always true? Is there any exceptions? This is the main discussion point here.

So, is this result always true? As long as we are in the range of secondary level, probably this is true. But if we go beyond that, the result may not hold good. Let’s see how.

To disprove any result in mathematics, one common strategy is to provide counter example. Let’s start with Matrices and then we will consider Vectors. Before we start our actual discussion, let us recall few important definitions. First one in our list is the zero (or null) matrix, denoted by O. If each element of an m × n matrix be 0, the matrix is said to be the zero matrix. Observe the difference between the real number 0 and the zero (or null) matrix O. Few null matrices of various orders are given below:

Is it always true that AB = 0 means either A = 0 or B = 0? (2)

We now consider three matrices A, B and C as given below:

Is it always true that AB = 0 means either A = 0 or B = 0? (3)

We can check the following two matrix multiplications (recall that, in case of matrix multiplications, AB means A multiplied by B and that the order is very important, so that AB and BA are different):

Is it always true that AB = 0 means either A = 0 or B = 0? (4)

From the matrix viewpoint, though AB = O or BC = O but we can see that none of A, B or C is zero! To be specific, none of them is a zero matrix.

Next, vectors are denoted by the bold face letters, like X. A zero (or null) vector (represented by P = 0) means a vector whose magnitude is zero, like |P| = 0. Now consider two vectors A and B and suppose θ is the angle between them. Their dot product is represented by A · B. Note that the dot product of two vectors is a scalar quantity. If their dot product is zero, then can we conclude A = B = 0? Let’s analyse.

Now A · B = 0 means |A||B| cos θ = 0. Three possibilities may come to our mind. They are

  • either |A| = 0;
  • or |B| = 0;
  • or cos θ = 0 (means θ = π/2)

In other words, if A = 0 then A · B = 0; if B = 0 then A · B = 0; even if θ = π/2 (means the two vectors A and B are orthogonal/ perpendicular) then A · B = 0. Hence A · B = 0 not always mean that A = 0 or B = 0.

Is it always true that AB = 0 means either A = 0 or B = 0? (5)

Next, we will consider their cross product namely A × B, which is a vector quantity. Point of discussion here is, if their cross product is zero can we conclude A = B = 0? Let’s analyse.

With usual notation, A × B = 0 means |A||B| sin θ n = 0, where n is a unit vector perpendicular (or normal) to the plane containing both A and B. Again, three possibilities we need to consider and they are

  • either |A| = 0;
  • or |B| = 0;
  • or sin θ = 0.

In other words, if A = 0 then A × B = 0; if B = 0 then A × B = 0; even if θ = 0 (means the vectors A and B are parallel) then also A × B = 0. Hence A × B = 0 not always mean that A = 0 or B = 0.

What follows from the above discussion is that, whenever we are concluding A = 0 or B = 0 or A = 0 = B from AB = 0, we need to be specific about A and B. Needless to say, if we fail to understand the underlying object we might conclude wrong. Above examples, definitely help us to understand this fact.

Your suggestions are eagerly and respectfully welcome!See you soon with a new mathematics blog that you and I callMath1089 – Mathematics for All!“.

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Is it always true that AB = 0 means either A = 0 or B = 0? (2024)

FAQs

Is it always true that AB = 0 means either A = 0 or B = 0? ›

The Zero Product Property states that if the product of two numbers is zero, then at least one of the numbers is zero. In symbols, where a and b represent numbers, if ab=0, then a = 0 or b=0.

Does AB 0 implies either a 0 or B 0? ›

The Zero Product Property states that if the product of two numbers is zero, then at least one of the numbers is zero. In symbols, where a and b represent numbers, if ab=0, then a = 0 or b=0.

How to prove if ab 0 then a 0 or b 0? ›

Observe that for the product to be zero at least one of the factors must be zero. In general: If a x b = 0 ⇒ a = 0 and/or b = 0.

Is true or false if ab is a zero matrix then either a or b is a zero matrix? ›

So, we can say that the product of two matrices can be the null matrix while neither of them is a null matrix. Hence, If the matrix AB is zero, then it is not necessary that either A = O or, B = O.

Does the zero product property simply states that if ab 0 then either a 0 or b 0 or both? ›

The Zero Product Property simply states that if a b = 0 , then either a = 0 or b = 0 (or both). A product of factors is zero if and only if one or more of the factors is zero. This is particularly useful when solving quadratic equations .

What is ab 0 only if? ›

a.b = 0 if any of the a and b or both will be 0. It means a.b <0 if any of the a and b is <0. It means for a.b = 0 a and b both must be either <0 or > 0.

What does ab 0 mean? ›

If A and B are vectors (say A is a row vector and B is column vector), then the dot product is 0. This means that the two vectors are orthogonal to one another. If A and B are matrices, AB = 0 means that the vectors in the columns of B are orthogonal to the space spanned by the rows of A.

How do you prove AxB )= 0? ›

The proof that A. (AxB)=0 lies in the understanding of vector relationships and mathematical operations. AxB results in a vector perpendicular to A and B. Therefore, A's dot product with a vector perpendicular to it will be zero as the cosine of 90 degrees is zero.

Can you have AxB AB with a not equal to zero? ›

A × B = A • B with A ≠ 0 and B ≠ 0This is because the left hand side of the given equation gives a vector quantity while the right hand side gives a scalar quantity. However if one of the two vectors is zero then both the sides will be equal to zero and the relation will be valid.

How do you tell if an equation is true false or open? ›

Equations can be true, false, or open. That is, if an equation is a true statement, then it is a true equation, if an equation is a false statement, then it is a false equation, and if an equation's truth value depends on the values of the variables in the equation, then we call it an open equation.

What is the condition for AB 0? ›

The condition is that the column vectors of B must be in the null space (aka kernel) of the matrix A, i.e., AX = 0 for each column X of B. There may be any number of such columns in B, and the resulting zero matrix AB = O will have #rows = #rows(A) and #columns = #cols(B).

How do you know if the inverse of a matrix does not exist? ›

If the determinant of the matrix is zero, then it will not have an inverse; the matrix is then said to be singular. Only non-singular matrices have inverses. Find the inverse of the matrix A = ( 3 1 4 2 ). result should be the identity matrix I = ( 1 0 0 1 ).

Does AB 0 prove a 0 or B 0? ›

Since ab = 0, it follows that a = 0 or b = 0. If a = 0, then a + b = b. Since a + b = 0, it then follows that b = 0. Hence it then follows that a = 0 and b = 0.

Which property states that if ab 0 then either a, b, or both must equal zero? ›

The Zero Product Property states that if the product of two numbers is zero, then at least one of the numbers is zero. In symbols, if ab=0, then a=0 or b=0.

What property always equals 0? ›

Zero property of multiplication is defined as “when we multiply any number by zero, the resulting product is always a zero”.

Can you have a → b → a → ⋅ b → with a ≠ 0 and b ≠ 0 what if one of the two vectors is zero? ›

This is because the left hand side of the given equation gives a vector quantity, while the right hand side gives a scalar quantity. However, if one of the two vectors is zero, then both the sides will be equal to zero and the relation will be valid.

What is it called for any real number a and b if ab 0 then either a 0 or b 0? ›

The Zero-Product Property states that for all real numbers a and b, if ab = 0 then either a = 0 or b =0.

Can A and B both be zero? ›

Notice that if both A and B are zero, then there is no way you could make such and expression.

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