Determine whether A and B are inverses by calculating AB and...

Determine whether A and B are inverses by calculating AB and BA. Do not use a calculator.
$$A=\left[\begin{array}{lll}
1 & 3 & 3 \\
1 & 4 & 3 \\
1 & 3 & 4
\end{array}\right] ; B=\left[\begin{array}{rrr}
7 & -3 & -3 \\
-1 & 1 & 0 \\
-1 & 0 & 1
\end{array}\right]$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Matrix Multiplication

Matrix multiplication is an operation where two matrices are combined to produce a third matrix. The entry in the ith row and jth column of the resulting matrix is obtained by taking the sum of the products of corresponding entries from the ith row of the first matrix and the jth column of the second matrix. This operation is not commutative, meaning that the order in which matrices are multiplied matters, and it is a fundamental tool in linear algebra for combining linear transformations.

Inverse Matrix

An inverse matrix is a matrix that, when multiplied with the original matrix, yields the identity matrix. For a square matrix A, its inverse, denoted by A?¹, exists if and only if A is nonsingular, that is, it has a nonzero determinant. This concept is analogous to the reciprocal of a number, and it is essential in solving systems of linear equations, among other applications in linear algebra.

Identity Matrix

The identity matrix is a special type of square matrix characterized by having ones on its main diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix multiplication, meaning that any matrix multiplied by the identity matrix (of compatible size) will yield the original matrix. In the context of inverse matrices, multiplying a matrix by its inverse should result in the identity matrix.

Please give Ace some feedback