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Chapter 12: Problem 4

If \(A\) is a matrix, state conditions on \(A\) for \(A^{2}\) to exist.

Short Answer

Expert verified
Answer: A matrix can be squared if it is a square matrix, meaning it has the same number of rows and columns (i.e., an \(n \times n\) matrix).

Step by step solution

01

Understanding matrix multiplication

Matrix multiplication is only possible if certain criteria are met. Specifically, given two matrices, \(A\) and \(B\), their product, \(AB\), is defined only if the number of columns in \(A\) is equal to the number of rows in \(B\). In other words, if \(A\) is an \(m \times n\) matrix and \(B\) is a \(p \times q\) matrix, then \(AB\) will exist if and only if \(n = p\).
02

Applying the matrix multiplication condition to a square matrix

Now we want to find conditions under which \(A^2\) exists. Recall that \(A^2\) is simply the product of \(A\) with itself, i.e., \(A \times A\). Let \(A\) be an \(m \times n\) matrix. In order for \(A^2\) to exist, the number of columns in \(A\) must be equal to the number of rows in \(A\). Using the condition outlined in Step 1, this means that \(n = m\).
03

Stating the condition for \(A^2\) to exist

Based on the discussion in Step 2, we can now state the condition required for \(A^2\) to exist: \(A\) must be a square matrix. In other words, \(A\) must have the same number of rows and columns (i.e., \(A\) must be an \(n \times n\) matrix).

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Most popular questions from this chapter

State the number of elements in an \(n \times m\) matrix.

Refer to matrices \(P, Q\) and \(R\) where \(P\) is a \(3 \times 2\) matrix, \(Q\) is a \(3 \times 3\) matrix and \(R\) is a \(2 \times 3\) matrix. State the size of the following products. If a product cannot be found then state this. (a) \(P Q R\) (b) \(P R Q\) (c) \(Q P R\) (d) \(R Q P\)

Given $$ A=\left(\begin{array}{cc} 2 & 1 \\ 3 & -2 \end{array}\right), \quad B=\left(\begin{array}{cc} 4 & -1 \\ -2 & 6 \end{array}\right) $$ find (a) \(3 A\) (b) \(2 B\) (c) \(4 A+3 B\) (d) \(B-2 A\) (e) \(2 A^{\mathrm{T}}\) (f) \((2 A)^{\mathrm{T}}\)

Given find \(R=\left(\begin{array}{ll}4 & 1 \\ 6 & 2\end{array}\right), \quad S=\left(\begin{array}{cc}-3 & 2 \\ 7 & -4\end{array}\right)\) (a) \(R+S\left(\right.\) b) \(S-R\) (c) \(R+R^{\mathrm{T}}\) (d) \((R-S)^{\mathrm{T}}\) (e) \(\left(S^{\mathrm{T}}\right)^{\mathrm{T}}\)

State the transpose of \(C\) where $$ C=\left(\begin{array}{cc} 9 & 3 \\ 1 & -2 \\ \alpha & 4 \end{array}\right) $$
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