The product of a matrix and its inverse equals which matrix?

The product of a matrix and its inverse equals which matrix?

• A. The identity matrix
• B. The zero matrix
• C. The original matrix
• D. The transpose

Answer: (A)

The main idea is that an inverse undoes the effect of a matrix under multiplication. For a square matrix A that has an inverse A^{-1}, multiplying them in either order gives the identity matrix I: A multiplied by A^{-1} equals I, and A^{-1} multiplied by A also equals I. The identity matrix acts like the number 1 does for ordinary multiplication: it leaves whatever it’s multiplied with unchanged, but here the result is not the original matrix because the inverse is specifically chosen to cancel A. That’s why the product is the identity matrix. The zero matrix isn’t possible here, because getting a zero result would imply A is not invertible (its determinant would be zero), which contradicts the existence of an inverse. The original matrix isn’t the result because the whole purpose of the inverse is to reverse A’s effect, yielding the identity. The transpose isn’t involved in this defining property; it has its own role and doesn’t generally produce the identity when multiplied by A.

The main idea is that an inverse undoes the effect of a matrix under multiplication. For a square matrix A that has an inverse A^{-1}, multiplying them in either order gives the identity matrix I: A multiplied by A^{-1} equals I, and A^{-1} multiplied by A also equals I. The identity matrix acts like the number 1 does for ordinary multiplication: it leaves whatever it’s multiplied with unchanged, but here the result is not the original matrix because the inverse is specifically chosen to cancel A. That’s why the product is the identity matrix.

The zero matrix isn’t possible here, because getting a zero result would imply A is not invertible (its determinant would be zero), which contradicts the existence of an inverse. The original matrix isn’t the result because the whole purpose of the inverse is to reverse A’s effect, yielding the identity. The transpose isn’t involved in this defining property; it has its own role and doesn’t generally produce the identity when multiplied by A.