Question

Choose the correct statement or statements:
A. Every scalar matrix is an identity matrix.
B. Every identity matrix is a scalar matrix.
C. Transpose of a transpose of a matrix gives the matrix itself.
D. For every square matrix A there exists another matrix B such that AB=I=AB

Answer 1

This question and its solution is only and only dependent on the concepts and definitions of the different types of matrices. We will just recall those definitions and will check for the correct and incorrect statements.

Complete step by step answer:
Now let's start with the definitions first.
Scalar matrix: A Scalar matrix is a matrix in which every non-diagonal element is zero (${a_{ij}} = 0,{\text{i}} \ne j$) and all diagonal elements are equal (${a_{ij}}{\text{ = }}k,i = j$).

Identity matrix: An Identity Matrix is a matrix in which every non-diagonal element is zero (${a_{ij}} = 0,{\text{i}} \ne j$) and every diagonal element is 1 (${a_{ij}}{\text{ = 1}},i = j$).

Square matrix: A Square Matrix is a matrix of order $m \times n$ , such that m=n.

Transpose of a matrix: The matrix obtained by interchanging the rows and columns of a matrix is called the transpose of a matrix. If $A = {\left[ {{a_{ij}}} \right]_{m \times n}}$, be a matrix of order $m \times n$. Then the transpose of that matrix is denoted by ${A^1}$ or ${A^T}$ and is obtained by interchanging the rows and columns.
${A^1} = {A^T} = {\left[ {{a_{ij}}} \right]_{n \times m}}$

Now we will move towards the statements: The first statement is every scalar matrix is an identity matrix is not correct because for a matrix to be identity the diagonal elements must be 1 only but in scalar they can be any number along with 1.
Let's say a matrix A is a scalar matrix.
$A = kB$
Where k is the scalar that is to be multiplied with the diagonal elements.
Whereas an identity matrix is I = \left[ {\begin{array}{*{20}{c}} 1&0 \\\ 0&1 \end{array}} \right] can be termed as scalar because the non-diagonal elements are zero.

In statement second, we can easily answer it that is correct one because for a matrix to be scalar there is condition that non diagonal elements should be zero and diagonal elements
Should be the same. And that is well satisfied by the identity matrix.

1&0 \\\ 0&1 \end{array}} \right]$$ is an identity matrix that can be termed as scalar because diagonal elements are 1 and the same. For the third statement we can say when a matrix is once transposed, we get rows and columns changed. But if one more time it is transposed, we will definitely get the original matrix. If $$A = \left[ {\begin{array}{*{20}{c}} 4&3 \\\ 8&{ - 6} \end{array}} \right]$$ if transposed we get, $${A^1} = \left[ {\begin{array}{*{20}{c}} 4&8 \\\ 3&{ - 6} \end{array}} \right]$$ now if one more time it is transposed, we get, $${\left( {{A^1}} \right)^1} = \left[ {\begin{array}{*{20}{c}} 4&3 \\\ 8&{ - 6} \end{array}} \right]$$ that is the original matrix. Thus this statement is true. In the fourth statement there is no specifically mentioned which type of matrix is matrix B.So we cannot conclude about the correctness and incorrectness of the statement. **Thus, statements B and C are correct.** **Note:** matrix is a very easy to understand and easy concept. Rather generally mistakes are done in easy things. In the fourth statement if they mention that matrix B is an inverse matrix of A then that statement is correct but not for any general matrix like B.Also statement C is the property of transposes of matrices. $${\left( {{A^1}} \right)^1} = A$$