An m x n matrix is invertible if and only if it is a square matrix (m = n) and its determinant is non-zero. In other words, for a matrix to be invertible, it must have the same number of rows and columns, and its determinant must not be equal to zero. If these conditions are met, then the matrix is invertible and has a unique inverse. If the matrix does not meet these conditions, then it is not invertible. **

When matrix A is multiplied by its transposed cofactor matrix C, the result is the determinant of matrix A times the identity matrix. This is because the product of a matrix and its transposed cofactor matrix is equal to the determinant of the original matrix times the identity matrix. In other words, the result is a scalar multiple of the identity matrix. **

An elliptical orbit is a type of orbit in which an object, such as a planet or satellite, follows an oval-shaped path around another object, typically a star or planet. In an elliptical orbit, the object's distance from the central body varies, with the closest point being the perigee and the farthest point being the apogee. The shape of the ellipse is determined by the gravitational pull of the central body and the object's velocity. Elliptical orbits are common in our solar system, with planets like Earth and Mars following elliptical paths around the Sun. **

Elliptical orbits occur due to the gravitational forces between celestial bodies. When an object is under the influence of a central force, such as gravity, its path will often take the shape of an ellipse. This is because the force of gravity is stronger at shorter distances, causing the object to move faster when closer to the center of mass and slower when farther away. As a result, the object follows a curved path that is elliptical in shape. **

The result of the multiplication of matrix A with its transposed cofactor matrix C is the determinant of matrix A times the identity matrix. In other words, if A is an n x n matrix, then A multiplied by the transposed cofactor matrix C will result in a matrix with diagonal elements equal to the determinant of A and all other elements equal to zero. This result is a consequence of the properties of the cofactor matrix and its relationship to the determinant of A. **

To create a small matrix in C, you can declare a 2D array with a fixed size. For example, to create a 3x3 matrix, you can declare an array like this: `int matrix[3][3];`. You can then access and manipulate the elements of the matrix using nested loops. Remember that in C, arrays are zero-indexed, so the first element of the matrix would be `matrix[0][0]`. **

The planet orbits are elliptical because of the gravitational force exerted by the sun. According to Kepler's laws of planetary motion, planets move in elliptical orbits with the sun at one of the foci. This means that the gravitational force between the sun and the planet causes the planet to move in a curved path, resulting in an elliptical orbit. The shape of the orbit is determined by the balance between the planet's velocity and the gravitational force pulling it towards the sun. **

To square a matrix in matrix algebra, you simply multiply the matrix by itself. This means you multiply the matrix by itself using matrix multiplication rules. The resulting matrix will be the square of the original matrix. It is important to ensure that the dimensions of the matrix allow for matrix multiplication, meaning the number of columns in the first matrix must be equal to the number of rows in the second matrix. **