11. Matrices and determinants

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Matrices and determinants are fundamental concepts of mathematics that are often required in the ENEM test. Understanding these concepts and knowing how to apply them can make a big difference to your final score.

Matrices

Matrices are a way of organizing data into rows and columns, forming a table. Each position in the array is called an element. The dimension of a matrix is ​​given by the number of rows and columns it has. For example, a 3x2 dimension matrix has 3 rows and 2 columns.

There are several types of matrices, including the row matrix (a single row), the column matrix (a single column), the square matrix (the number of rows equals the number of columns), the diagonal matrix (all elements off the main diagonal are zero), the identity matrix (a diagonal matrix where all elements on the main diagonal are one), and the null matrix (all elements are zero).

Operations with matrices include addition and subtraction (adding or subtracting the corresponding elements of two matrices), multiplication by a scalar (multiplying all elements of the matrix by a number), and matrix multiplication (a more complex operation which involves multiplying the elements of a row of the first matrix by the elements of a column of the second matrix and adding the results).

Determinants

The determinant is a special value that can only be calculated for square matrices. It has several important properties and applications, including solving systems of linear equations, inverting matrices, and finding the area of ​​a triangle in a coordinate plane.

To calculate the determinant of a 2x2 matrix, you multiply the elements on the main diagonal and subtract the product of the elements on the secondary diagonal. For a 3x3 matrix, the calculation is more complex and involves the creation of "minors" and "cofactors".

Determinants also have their own properties, including the fact that the determinant of a matrix is ​​equal to the determinant of its transpose, the determinant of a matrix multiplied by a scalar is equal to the scalar times the determinant of the matrix, and the The determinant of the product of two matrices is equal to the product of the determinants of the matrices.

In summary, matrices and determinants are powerful mathematical tools that allow you to organize data and perform complex calculations in a systematic and orderly way. Mastering these concepts is essential for any student who wants to get a good score on the ENEM.

To prepare for the test, it is recommended that students practice solving problems involving matrices and determinants, review the properties of these concepts, and become familiar with the different operations that can be performed with them. Also, it's important to understand how these concepts are applied in real contexts, as this can help answer multiple-choice questions and solve problems more efficiently.

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