Investigating the Properties of Matrix Multiplication: False Statement, Matrix AB = 0

If A and B are matrices, and the product AB is the zero matrix, then A or B must be a zero matrix.

To determine if the statement “If A and B are matrices, and the product AB is the zero matrix, then A or B must be a zero matrix” is true or false, we need to investigate the properties of matrix multiplication

To determine if the statement “If A and B are matrices, and the product AB is the zero matrix, then A or B must be a zero matrix” is true or false, we need to investigate the properties of matrix multiplication.

In general, for two matrices A and B to be multiplied, the number of columns in A must be equal to the number of rows in B. If the product AB is defined, it will result in a new matrix with the same number of rows as A and the same number of columns as B.

Let’s assume that A and B are matrices of appropriate sizes for multiplication, and the product AB is the zero matrix. Mathematically, we can write this as AB = 0, where 0 represents the zero matrix.

Now, if AB = 0, it does not necessarily imply that either A or B is a zero matrix. It is entirely possible to have non-zero matrices A and B such that their product yields the zero matrix.

To illustrate this, consider the following example:

A = [1 0]
[0 0]

B = [0 0]
[0 1]

In this example, both A and B are non-zero matrices, but when we multiply them, we get:

AB = [1*0 + 0*0 1*0 + 0*1]
[0*0 + 0*0 0*0 + 0*1]

Simplifying the above expression, we have:

AB = [0 0]
[0 0]

As we can see, the product AB is the zero matrix, but neither A nor B individually is a zero matrix.

Hence, the statement “If A and B are matrices, and the product AB is the zero matrix, then A or B must be a zero matrix” is FALSE.

More Answers:

The Relationship Between Coincident Lines and Infinitely Many Solutions in a System of Linear Equations
Proving that if AB = 0, then either A or B is a zero matrix
Proving C = D: All Corresponding Entries in C and D are Equal

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