Question

There exists a $\mathbf{4}\mathbf{\times }\mathbf{4}$ matrix whose entries are all 1or -1 , and such that$\mathit{d}\mathit{e}\mathit{t}\mathit{A}\mathbf{=}\mathbf{16}$.

Short answer

Therefore, the given statement is false.

Step-by-step solution

Matrix Definition

A matrix is aset of numbers arranged in rows and columns so as to form a rectangulararray.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an “m by n ” matrix, written “$m\times n$ .”

To check whether the given condition is true or false

We know that determinant of similar matrices is equal.

So, det (A) must be equal to, det (2A)which is false.

If A is a $4\times 4$matrix with entries -1 or 1 , then all 24 of the addends in the definition of can be either - 1 or 1 .

If we want det A= 16, then twenty of them must be 1, and the remaining four be -1, which is impossible.