Therefore, given a matrix A = (theij)mxn the transpose of A is At = (a 'ji) nxm.

i: position on the line
j: column position
unij: an array element at position ij
m: number of rows in the matrix
n: number of columns in the matrix
Unt: transpose matrix of A

Note that matrix A is of order mxn, while its transpose At is of order nx m.

Example

Find the transpose matrix of matrix B.

As the given matrix is ​​of type 3 × 2 (3 rows and 2 columns), its transposition will be of type 2 × 3 (2 rows and 3 columns).
To construct the transpose matrix, we must write all the columns of B as lines of Bt. As indicated in the following diagram:

Therefore, the transpose matrix of B will be:

## Properties of the transpose matrix

• (At)t = A: this property indicates that the transpose of a transposed matrix is ​​the original matrix.
• (A + B)t = At + Bt: the transposition of the sum of two matrices is equal to the sum of the transposition of each of them.
• (A.B)t =Bt . at: the transposition of the multiplication of two matrices is equal to the product of the transpositions of each of them, in reverse order.
• det(M) = det(Mt): the determinant of the transposed matrix is ​​the same as the determinant of the original matrix.

## Symmetric matrix

A matrix is ​​called symmetric when, for any element of matrix A, the equality aij = Aji It is true

Matrices of this type are square matrices, that is, the number of rows is equal to the number of columns.

Each symmetric matrix satisfies the following relationship:

A = At

## Opposite matrix

It is important not to confuse the opposite matrix with the transpose one. The opposite matrix is ​​one that contains the same elements in rows and columns, however, with different signs. Therefore, the opposite of B is –B.

## Inverse matrix

The inverse matrix (indicated by the number –1) is one in which the product of two matrices is equal to a square identity matrix (I) of the same order.

Example:

A. B = B. A = In (when matrix B is inverse of matrix A)

## Vestibular exercises with feedback

1. (Fei-SP) Given the matrix A =, where At its transposition, the determinant of matrix A. At is

a) 1
b) 7
c) 14
d) 49

2. (FGV-SP) A and B are headquarters and At is the transpose matrix of A. If, then matrix At . B will be null for:

a) x + y = –3
b) x. y = 2
c) x / y = –4
d) x. and2 = –1
e) x / y = –8

3. (UFSM-RS) Knowing that the matrix

equals transpose, the value of 2x + y is:

a) –23
b) –11
c) –1
d) 11
e) 23