A triangle with sides 3 cm, 4 cm, and 5 cm satisfies the Pythagorean theorem (3² + 4² = 5²), thus it is a right triangle.

The area of a circle is given by A = πr². If the radius is doubled (r becomes 2r), the new area is A' = π(2r)² = 4πr², which is four times the original area.

The determinant of an upper triangular matrix is the product of its diagonal elements: 1 * 1 * 1 = 1.

Det(G) = 0 because the first column has a zero entry, leading to a linear dependence.

In a parallelogram, opposite angles are equal. Therefore, if one angle is 70 degrees, the opposite angle is also 70 degrees.

In a 30-60-90 triangle, the sides opposite the 30 degrees and 60 degrees angles are in the ratio 1:√3.

The area of a triangle is given by A = 1/2 * base * height. Here, A = 1/2 * 8 * 5 = 20 cm².

The characteristic polynomial is det(E - λI) = det([[2-λ, 1], [1, 2-λ]]) = (2-λ)(2-λ) - 1 = λ^2 - 3λ + 1.

The characteristic polynomial is given by det(G - λI) = det([[2-λ, 1], [1, 2-λ]]) = (2-λ)(2-λ) - 1 = λ^2 - 3λ + 3.

The circumference C of a circle is given by C = π * diameter. Here, C = π * 14 cm ≈ 3.14 * 14 ≈ 44 cm.

The determinant of a 1x1 matrix is simply the value of the single element. Therefore, the determinant of [[5]] is 5.

The determinant of a 2x2 matrix is calculated as ad - bc.

The determinant of a 2x2 matrix is calculated as ad - bc.

The determinant of E is 0 because the rows are linearly dependent.

The determinant can be calculated using the formula for 3x3 matrices. Here, the first column is the same, leading to a determinant of 0.

The determinant is 0 because the first column is a linear combination of the other columns.

Det(J) = (5*8) - (6*7) = 40 - 42 = -2.

The length of the diagonal d of a rectangle can be found using the Pythagorean theorem: d = √(length² + width²) = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.

The order of a matrix is given by the number of rows followed by the number of columns. Therefore, a matrix with 3 rows and 4 columns is of order 3x4.

The perimeter of a rectangle is given by P = 2(length + width). Here, P = 2(10 + 5) = 30 cm.

The rows of B are linearly dependent, hence the rank of B is 2.

The rows of D are linearly dependent, hence the rank of D is 2.

The rank of a matrix is the maximum number of linearly independent row vectors. Here, the first and third rows are independent, while the second row is zero. Thus, the rank is 2.

The rank of E is 2 because the rows are linearly dependent (the third row is a linear combination of the first two).

In a rectangle, the diagonals are equal in length.

The trace of a matrix is the sum of its diagonal elements. For matrix A, the trace is 3 + 4 = 7.

The trace of a matrix is the sum of its diagonal elements. For matrix A, the trace is a + d.

The trace of a matrix is the sum of its diagonal elements. For the matrix [[a, b], [c, d]], the trace is a + d.

The trace of a matrix is defined as the sum of the elements on the main diagonal.

The trace of a square matrix is defined as the sum of the elements on its main diagonal.