Matrices And Determinants Class 11 Solutions Pdf Download

Here, in these NCERT Solutions, you will get a complete description of applications of determinants and matrices for solving the system of linear equations in two or three variables and how to check the consistency of the system of linear equations.

In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants.

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The determinant of an n n matrix can be defined in several equivalent ways, the most common being Leibniz formula, which expresses the determinant as a sum of n ! {\displaystyle n!} (the factorial of n) signed products of matrix entries. It can be computed by the Laplace expansion, which expresses the determinant as a linear combination of determinants of submatrices, or with Gaussian elimination, which expresses the determinant as the product of the diagonal entries of a diagonal matrix that is obtained by a succession of elementary row operations.

The determinant has several key properties that can be proved by direct evaluation of the definition for 2 2 {\displaystyle 2\times 2} -matrices, and that continue to hold for determinants of larger matrices. They are as follows:[1] first, the determinant of the identity matrix ( 1 0 0 1 ) {\displaystyle {\begin{pmatrix}1&0\\0&1\end{pmatrix}}} is 1.Second, the determinant is zero if two rows are the same:

There are various equivalent ways to define the determinant of a square matrix A, i.e. one with the same number of rows and columns: the determinant can be defined via the Leibniz formula, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending on the entries of the matrix satisfying certain properties. This approach can also be used to compute determinants by simplifying the matrices in question.

To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a standard basis vector. These determinants are either 0 (by property 9) or else 1 (by properties 1 and 12 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of an appropriate function is not clear.[citation needed]

These characterizing properties and their consequences listed above are both theoretically significant, but can also be used to compute determinants for concrete matrices. In fact, Gaussian elimination can be applied to bring any matrix into upper triangular form, and the steps in this algorithm affect the determinant in a controlled way. The following concrete example illustrates the computation of the determinant of the matrix A {\displaystyle A} using that method:

The determinant is a multiplicative map, i.e., for square matrices A {\displaystyle A} and B {\displaystyle B} of equal size, the determinant of a matrix product equals the product of their determinants:

Unwinding the determinants of these 2 2 {\displaystyle 2\times 2} -matrices gives back the Leibniz formula mentioned above. Similarly, the Laplace expansion along the j {\displaystyle j} -th column is the equality

Historically, determinants were used long before matrices: A determinant was originally defined as a property of a system of linear equations.The determinant "determines" whether the system has a unique solution (which occurs precisely if the determinant is non-zero).In this sense, determinants were first used in the Chinese mathematics textbook The Nine Chapters on the Mathematical Art (, Chinese scholars, around the 3rd century BCE). In Europe, solutions of linear systems of two equations were expressed by Cardano in 1545 by a determinant-like entity.[22]

For square matrices with entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be given to the Leibniz formula provided that the order for the product is specified, and similarly for other definitions of the determinant, but non-commutativity then leads to the loss of many fundamental properties of the determinant, such as the multiplicative property or that the determinant is unchanged under transposition of the matrix. Over non-commutative rings, there is no reasonable notion of a multilinear form (existence of a nonzero bilinear form[clarify] with a regular element of R as value on some pair of arguments implies that R is commutative). Nevertheless, various notions of non-commutative determinant have been formulated that preserve some of the properties of determinants, notably quasideterminants and the Dieudonn determinant. For some classes of matrices with non-commutative elements, one can define the determinant and prove linear algebra theorems that are very similar to their commutative analogs. Examples include the q-determinant on quantum groups, the Capelli determinant on Capelli matrices, and the Berezinian on supermatrices (i.e., matrices whose entries are elements of Z 2 {\displaystyle \mathbb {Z} _{2}} -graded rings).[49] Manin matrices form the class closest to matrices with commutative elements.

While the determinant can be computed directly using the Leibniz rule this approach is extremely inefficient for large matrices, since that formula requires calculating n ! {\displaystyle n!} ( n {\displaystyle n} factorial) products for an n n {\displaystyle n\times n} -matrix. Thus, the number of required operations grows very quickly: it is of order n ! {\displaystyle n!} . The Laplace expansion is similarly inefficient. Therefore, more involved techniques have been developed for calculating determinants.

Indeed. If you think that determinants are taught in order to invert matrices and compute eigenvalues, it becomes clear very soon that Gaussian elimination outperforms determinants in all but the smallest instances.

The Schur functions were defined using determinants. There is the classical definition as a determinant divided by the Vandermonde determinant. There is also the Jacobi-Trudi formula which expresses the Schur function as a determinant of complete homogeneous functions (or elementary functions). This came well before any theory of linear algebra.

At least on the numerical front: the computation of determinants is prone to overflow for large enough matrices, which is why libraries like LINPACK made provisions for separately computing the mantissa and the exponent (see this for instance).

Another application I have seen for determinants is as a check for the positive definiteness of a symmetric matrix through computing successive determinants of leading submatrices (this has applications in signal processing for instance); this too is slow compared to using e.g. Cholesky decomposition for checking if a matrix is positive definite.

1) The Chern-Weil theory of characteristic classes is built upon determinants of functions of curvature forms of vector bundles.2) Feynman path integrals require determinants (but typically in infinite dimensions).

But there is a condition to obtain a matrix determinant, the matrix must be a square matrix in order to calculate it. Hence, the simplified definition is that the determinant is a value that can be computed from a square matrix to aid in the resolution of linear equation systems associated with such matrix. The determinant of a non square matrix does not exist, only determinants of square matrices are defined mathematically.

The determinant of a matrix can be denoted simply as det A, det(A) or |A|. This last notation comes from the notation we directly apply to the matrix we are obtaining the determinant of. In other words, we usually write down matrices and their determinants in a very similar way:

This equation is also known as the formula of determinant for a 3x3 matrix. At this point you may have noticed that finding the determinant of a matrix larger than 2x2 becomes a long ordeal, but the logic behind the process remains the same and so the difficulty is similar, the only key point is to keep track of the operations you are working through, even more with even larger matrices than a 3x3. In fact, the formula for determinants larger than 3x3 are even more atrocious, and are not worth memorizing.

The determinant of a 3x3 matrix shortcut method is a clever trick which facilitates the computation of a determinant of a large matrix by directly multiplying and adding (or subtracting) all of the elements in their necessary fashion, without having to pass through the matrix expansion of the first row and without having to evaluate secondary matrices' determinants.

In the last section of this lesson we will work through a set of three different 3x3 matrices and their determinants, we recommend you to compared the processes for both methods to understand them better.

Reciprocal of a matrix is called the Inverse of a matrix. Only square matrices with non-zero determinants are invertible. Suppose for any square matrix A with inverse matrix B their product is always an identity matrix (I) of the same order.

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