Statement 1 If A+B+C=π, then the value of

Key Concepts

Skew Symmetric Matrices

A skew symmetric matrix is one where the transpose of the matrix equals the negation of the matrix (A^T = -A). These matrices have unique properties, including that all its diagonal elements are zero. They appear frequently in various areas of linear algebra and its applications in physics, especially in the context of rotations and angular momentum.

Matrix Powers

Matrix powers involve multiplying a matrix by itself repeatedly. For skew symmetric matrices, an important observation is that odd powers retain the skew symmetric property, while even powers result in symmetric matrices. This distinction is critical when analyzing structural properties in algebraic expressions involving the powers of matrices.

Trigonometric Identities

Trigonometric identities relate the angles and sides of triangles through sine, cosine, tangent, and other functions. In problems where angles sum to a specific value (for example, ?), these identities can simplify expressions, such as knowing that sin(?) equals zero, which might reduce or nullify certain terms in an expression.

Determinant Properties

Determinants provide crucial information about a matrix, such as whether it is singular or non-singular. In matrices that include trigonometric functions, common simplifications such as zero entries or dependencies among rows and columns can directly lead to a determinant of zero, especially if one row or column becomes identically zero due to an identity or a constraint.

Transcript

00:01 So in the given question we have two statements that are given in the given and the first statement says that if a plus b plus c is equal to pi then in the determinant sign of a plus b plus c and sign b sign c and we have minus sign b 0 tan a and cost of a plus b minus tan a 0 the value of this determinant is equal to 0 so this is what is given as statement 1 in the questing right and in statement 2 it is told that every positive every positive integral power every positive integral power of a skew symmetric matrix, a skew symmetric matrix, skew symmetric matrix is also a skeu symmetric matrix.

01:35 So this is what is given as the second statement, right? and what we should do over here is to find out which of these statements are true and which one is false.

01:49 Or we can find whether both these statements are true and that is what we need to do.

01:58 So first let's look at this statement 2 which says every positive integral power of a skew symmetric matrix is skew symmetric.

02:09 So the property of a skew symmetric matrix is that when we take the transpose of a skew symmetric matrix we get minus a right so when we take the a positive integral power of these skew symmetric matrix a and take the transpose of this matrix what you would get is if the power of this matrix if the power of this matrix is an even power it would be a to the power n the transpose of this matrix would be be a to the power of n if n is a1 and we would get the answer as negative a to the power n if n is odd right so we get a skew symmetric matrix only when only when we take a matrix and take a positive integral power of a matrix a and find its transpose it only gives us the negative of a to the power n when n is owed, right? so the statement say, the statement two says that for every positive integral power of a skew symmetric matrix, we get a skew symmetric matrix, which is false because we only get a skew symmetric matrix for odd positive integral powers of a skew symmetric matrix, right? so then we can say statement two is a false statement.

03:47 Now let's look at the statement one, right? and in statement one, we are told that if a plus b plus c is equal to pi, then the value of this determinant would be equal to zero.

04:01 So a typical format of a skew symmetric matrix is of the form...

Please give Ace some feedback