Exercise 3.3 · Q1: Transposes of three matrices

Basics you need

Part (i): a column of three numbers

1. (i) State the order

   $$A = \begin{bmatrix} 5 \\ \tfrac12 \\ -1 \end{bmatrix}$$

   $A$ has 3 rows and 1 column, so it is a $3\times1$ matrix (a column matrix). Its entries are $a_{11} = 5$⁠, $a_{21} = \tfrac12$⁠, $a_{31} = -1$⁠. Its transpose will have order $1\times3$⁠: one row of three numbers.

   Knowing the order of the answer before you start is a quick guard against mistakes: if your answer does not have 1 row and 3 columns, something has gone wrong.
   Order is always "rows × columns" (Basic 1). Basics: order of a matrix

2. (i) Turn the rows into columns

   $A$ has three rows, each holding one number: $[5]$⁠, $[\tfrac12]$⁠, $[-1]$⁠. Writing row 1 as column 1, row 2 as column 2 and row 3 as column 3 gives

   $$A' = \begin{bmatrix} 5 & \tfrac12 & -1 \end{bmatrix}_{1\times3}.$$

   Check with the rule: $(A')_{12} = a_{21} = \tfrac12$ and $(A')_{13} = a_{31} = -1$⁠. ✓

   For a single column, "rows become columns" simply lays the column on its side. The order of the numbers is kept: the top entry becomes the leftmost entry. That is why the answer reads 5, then $\tfrac12$⁠, then $-1$⁠.
   $(A')_{ij} = a_{ji}$ (Basic 2). Lesson: transpose

Part (ii): a 2 × 2 matrix

1. (ii) State the order and the entries

   $$A = \begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}$$

   This is a $2\times2$ matrix, so $A'$ is also $2\times2$⁠. The entries are $a_{11} = 1$⁠, $a_{12} = -1$⁠, $a_{21} = 2$⁠, $a_{22} = 3$⁠.

   A square matrix keeps its order under transposing, because $n\times n$ turned round is still $n\times n$⁠. Only the entries move.

2. (ii) Write each row as a column

   Row 1 of $A$ is $1,\ -1$⁠; it becomes column 1 of $A'$⁠. Row 2 is $2,\ 3$⁠; it becomes column 2:

   $$A' = \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}.$$

   Entry check: $(A')_{12} = a_{21} = 2$ and $(A')_{21} = a_{12} = -1$⁠; the diagonal entries 1 and 3 do not move. ✓

   In a $2\times2$ matrix, transposing just swaps the two off-diagonal entries $a_{12}$ and $a_{21}$⁠. The entries on the main diagonal have equal row and column numbers, so swapping $i$ and $j$ leaves them in place.
   Entries $a_{ii}$ sit on the main diagonal and are unchanged by transposing. Basics: mirror patterns

Part (iii): a 3 × 3 matrix

1. (iii) State the order and read the rows

   $$A = \begin{bmatrix} -1 & 5 & 6 \\ \sqrt3 & 5 & 6 \\ 2 & 3 & -1 \end{bmatrix}$$

   This is $3\times3$⁠, so $A'$ is $3\times3$⁠. The three rows are

   • row 1: $-1,\ 5,\ 6$
   • row 2: $\sqrt3,\ 5,\ 6$
   • row 3: $2,\ 3,\ -1$
   Listing the rows separately makes the next step mechanical: each list is copied, in the same order, down one column.

2. (iii) Write row $k$ as column $k$

   Row 1 goes down column 1, row 2 down column 2, row 3 down column 3:

   $$A' = \begin{bmatrix} -1 & \sqrt3 & 2 \\ 5 & 5 & 3 \\ 6 & 6 & -1 \end{bmatrix}.$$

   Spot checks with $(A')_{ij} = a_{ji}$⁠:

   • $(A')_{12} = a_{21} = \sqrt3$
   • $(A')_{13} = a_{31} = 2$
   • $(A')_{23} = a_{32} = 3$
   • $(A')_{31} = a_{13} = 6$

   The diagonal $-1,\ 5,\ -1$ is unchanged. ✓

   The rule $(A')_{ij} = a_{ji}$ is the precise version of "rows become columns". Checking off-diagonal entries matters most, because those are the ones that move. Note that $a_{21} = \sqrt3$ and $a_{12} = 5$ trade places, so $\sqrt3$ ends up in row 1.
   Transpose: $A = [a_{ij}]_{m\times n} \Rightarrow A' = [a_{ji}]_{n\times m}$⁠, Definition 3. Lesson: transpose

3. Check by transposing back

   Transposing each answer again must return the matrix in the question, since $(A')' = A$⁠. For (iii), the rows of $A'$ are $-1, \sqrt3, 2$⁠; $5, 5, 3$⁠; $6, 6, -1$⁠, and writing them as columns gives back the original rows $-1, 5, 6$⁠; $\sqrt3, 5, 6$⁠; $2, 3, -1$⁠. ✓ The same works for (i) and (ii).

   A check that uses a different property from the one used to compute the answer catches copying slips. Since none of the three matrices equals its own transpose, none of them is symmetric; we meet that idea in Question 7.
   $(A')' = A$⁠, property (i) of Section 3.5.1 (Basic 3).

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