# Matrix Transpose Calculator

Created by Maciej Kowalski, PhD candidate
Reviewed by Anna Szczepanek, PhD and Jack Bowater
Last updated: Sep 26, 2022

Welcome to the matrix transpose calculator, where you'll have the opportunity to learn all about transposing matrices.

The determinant calculator and the inverse matrix calculator left us to know two of the most basic matrix operations: the determinant and the inverse. This calculator deals with a third basic matrix operation called transposition.

But we've all come here not just to see how to transpose a matrix but also to get the lowdown on all the matrix transpose properties. Don't worry; we'll go through all the information you need together before we even come to what is, say, the transpose of a 2x2 matrix.

Grab your morning/afternoon tea, and let's get going, shall we?

## What is a matrix?

What does "two" mean? What does it represent? Believe it or not, there is a school of thought called the philosophy of mathematics, where a couple of elder scientists ask these fundamental questions. Fortunately, we can leave them sitting there with their heads in the clouds while we deal with the problem more down-to-earthly.

Numbers, as we know, describe lengths, dimensions, weights, etc. For example, a fraction can be a pizza slice. A negative number can be a bank debt. Our point is that all these numbers, which we call real numbers, are... well, real. They appear all around us. Even the square root of a number describes the diagonal of a rectangle, and the famous π is present in all circle operations like those of our circle calc. So it seems like we know all there is to know about numbers, doesn't it?

Wrong. The real numbers are only where the fun starts. Fortunately, we're not here to go into the crazy (but very useful and exciting) extension called complex numbers. We'd like to go in a different direction: to try to understand objects we can't accurately describe with a single number.

A matrix is an array of elements (usually numbers) with a set number of rows and columns. An example of a matrix would be

$A = \begin{bmatrix} 3 & -1 \\ 0 & 2 \\ 1 & -1 \end{bmatrix}$

Moreover, we say that a matrix has cells, or boxes, into which we write the elements of our array. For example, the above matrix $A$ has the value $2$ in the cell located in the second row and the second column.

As you can see, matrices are a tool used to write a few numbers concisely and operate on the whole lot as a single object. As such, they are extremely useful when dealing with:

• systems of equations, especially when trying to find the reduced row echelon form of a system;
• vectors and vector spaces;
• 3-dimensional geometry (e.g., what we do in the dot product and the cross product calculator);
• eigenvalues and eigenvectors; and
• graph theory and discrete mathematics.

To understand matrices better, mathematicians define several operations. For example, we can add the arrays together. We can even multiply them, although this can get a little tricky. And when you want to find the inverse of a matrix via the so-called adjoint matrix, you need to know how to transpose a matrix (the cofactor matrix in this case), which brings us to why we're here right now.

## How to transpose a matrix?

To transpose a matrix, we just flip its cells so that what was a row before will now be a column and vice versa. To help visualize it, imagine that your matrix is written on a piece of paper. To find its transpose, it's enough to flip the sheet to the other side and rotate it 90 degrees counterclockwise.

To those who like formulas, let's say that we have a matrix $A$ whose cells are indexed by $a_{ij}$, where $i$ denotes the number of the row, and $j$ is the number of the column. For example, the element $a_{23}$ is in the second row of the third column. Then, the transpose of $A$, denoted by $A^T$, will have elements $a_{ji}$, i.e., the second index is now the number of the row, and the first is the number of the column. In particular, the $a_{23}$ from before will now be in the third row of the second column.

Easy enough, isn't it? Well, it's not rocket science, so let's stop at that. And even better - there are several interesting and useful matrix transpose properties!

## Matrix transpose properties

Now that we know how to transpose a matrix let's take a closer look at and try to find some matrix transpose properties that can simplify our calculations.

1. In general, the shape of the transpose is different from that of the original matrix. Say that we start with a matrix with $n$ rows and $m$ columns. After transposing, the first row (which has $m$ elements) will become the first column, which means the matrix transpose will have $m$ rows. Similarly, it will have $n$ columns (as opposed to $m$ in the initial matrix). Of course, the shape would not change only if our array was a square, for example, if we wanted to find the transpose of a 2x2 matrix.

2. The transpose of a transpose is the initial matrix. By definition, calculating a matrix transpose means exchanging rows with columns. Therefore, if we do this flip a second time, we'll change them back and obtain what we started with. Symbolically, this property can be written as $(A^T)^T = A$, where $A$ is any matrix.

3. The transpose of a sum is the sum of the transposes. In other words, if we find ourselves adding (or subtracting) some matrices, it doesn't matter if we add them together first and transpose the result, or transpose each summand and then add them together. Symbolically, this means that $(A + B)^T = A^T + B^T$, where $A$ and $B$ are arbitrary matrices of the same size.

4. The transpose of a product is the product of the transposes reversed. This means that if we want to multiply two matrices and transpose the result, we can alternatively begin by transposing the factors and multiplying them but in reverse order. In other words, $(AB)^T = B^TA^T$ for any two matrices $A$ and $B$, for which $AB$ exists.

5. The determinant stays the same after transposition. If we have a square matrix and want to find its determinant, then we automatically know that its transpose will have the same determinant. Symbolically, this means that $|A| = |A^T|$ for any square matrix $A$ (here $| \ |$ denote the determinant, not the absolute value of a number.

All this time spent reading through the theory should be rewarded with a nice example, wouldn't you say?

## Example: using the matrix transpose calculator

Recall the matrix $A$ that we met :

$A = \begin{bmatrix} 3 & -1 \\ 0 & 2 \\ 1 & -1 \end{bmatrix}$

Let's use the matrix transpose calculator to find $A^T$.

The first thing we have to do is figure out the size. Our $A$ has three rows and two columns, so we need to tell that to our calculator by choosing the correct values under "number of rows" and "number of columns." This will show us a symbolic picture of a matrix similar to ours, with its cells denoted by $a_1$, $a_2$, $b_1$, and so on. We then have to input the data from $A$, i.e., the numbers in its boxes, as the symbols corresponding to the cells in the picture. Since our matrix has elements $3$ and $-1$ in its first row, and the picture tells us that the first row has symbols $a_1$ and $a_2$, we should write

$a_1 = 3$ and $a_2 = -1$.

Similarly, we fill out the other rows:

$b_1 = 0$, $b_2 = 2$,

$c_1 = 1$, and $c_2 = -1$.

This will make the matrix transpose calculator spit out the result. Nevertheless, let's try to also calculate the answer by hand.

The matrix $A$ has three rows and two columns, so its transpose $A^T$ will have a different shape: two rows and three columns. (Recall that for square matrices, the shape doesn't change, say, for the transpose of a 2x2 matrix.) Also, $A$'s first row has elements $3$ and $-1$, so we copy them into the first column of $A^T$:

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

Similarly, we put the second row of $A$ into the second column of $A^T$, and the third row into the third column.

$A^T = \begin{bmatrix} 3 & 0 & 1\\ -1 & 2 & -1\\ \end{bmatrix}$

That wasn't so bad, was it? No need to lose any more sleep on all those crucial matrix transpose questions. Omni Calculator saved the day yet again!

Maciej Kowalski, PhD candidate
Matrix size
Number of rows
2
Number of columns
2
A=
 ⌈ a1 a2 ⌉ ⌊ b1 b2 ⌋
Matrix entries
a₁
a₂
b₁
b₂
 ⌈ ⌉ᵀ = ⌈ ⌉ ⌊ ⌋ ⌊ ⌋
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