Question

Find (a) \(|A|\), (b) \(|B|\), (c) \(A B\), and (d) \(|A B|\). $$ \left[\begin{array}{rrr} 6 & -5 & 2 \\ 0 & 5 & -3 \\ 0 & 0 & 2 \end{array}\right] $$

Short answer

(a) \(|A|=60\), (b) \(|B|=1/60\), (c) \(AB=I\), (d) \(|AB|=1\)

Step-by-step solution

Calculate |A|

The determinant of a triangular matrix is the product of the entries on its main diagonal. Hence, \(|A|=6\times5\times2=60\)

Calculate |B|

Since B is the inverse of A and the determinant of an inverse matrix B is 1 divided by the determinant of the original matrix A, we have \(|B|=1/|A|=1/60\)

Calculate AB

Since B is the inverse of A, their product \(AB\) is the identity matrix I, which is \[\begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}\]

Calculate |AB|

The determinant of the identity matrix is always 1, hence \(|AB|=1\)

Key concepts

Triangular Matrix Determinant

In the realm of linear algebra, the determinant of a matrix is a special number that conveys significant properties of the matrix, including the ability to find its inverse, determine its rank, and solve systems of linear equations. When dealing with a triangular matrix, which is a type of square matrix where all the entries above or below the main diagonal are zero, calculating the determinant becomes particularly straightforward.

A triangular matrix looks like this for an upper triangular matrix:
\begin{bmatrix}a & b & c \ 0 & d & e \ 0 & 0 & f dmatrix}
Or like this for a lower triangular matrix:
\begin{bmatrix}a & 0 & 0 \ b & d & 0 \ c & e & f dmatrix}

The determinant of a triangular matrix, whether upper or lower, is simply the product of the elements along its main diagonal. This can be formally expressed as:\[|A| = a_{11} \times a_{22} \times ldots \times a_{nn}\]where \(a_{nn}\) are the diagonal elements of the matrix. This simplifies the often complex process of determinant calculation to a mere multiplication problem, as demonstrated in the exercise provided where \(|A| = 6 \times 5 \times 2 = 60\).

Inverse Matrix Determinant

An inverse of a matrix is analogous to the reciprocal of a number. Just as multiplying a number by its reciprocal yields the identity (or '1' for numbers), multiplying a matrix by its inverse will result in the identity matrix. The identity matrix is the equivalent of '1' for matrices, with ones on its diagonal and zeroes everywhere else.

Understanding the determinant of an inverse matrix requires one to dive a little into the properties of determinants themselves. In general, the determinant of an inverse matrix \(B = A^{-1}\), is the reciprocal of the determinant of the original matrix \(A\). It can be mathematically represented as:\[|B| = 1/|A|\]This relationship is crucial when considering the properties of matrix operations. As seen in the exercise example, if matrix \(A\) has a determinant of 60, the determinant of its inverse, \(B\), will be \(1/60\). The simplicity of this calculation again boils down the complexity of matrix algebra to a basic arithmetic operation.

Identity Matrix

The identity matrix plays a key role in matrix algebra. It is the matrix equivalent of the number '1' in regular multiplication. An identity matrix is a square matrix with ones on the diagonal and zeroes elsewhere. No matter what matrix \(A\) you multiply the identity matrix by, \(A\) will always remain unchanged, just as any number multiplied by one remains the same.

In terms of determinants, the identity matrix is remarkable for its simplicity:\[|I| = 1\]The determinant of any identity matrix, regardless of its size, will always be '1'. This property is illustrated in the solution to the exercise where after multiplying matrix \(A\) by its inverse \(B\), the result is the identity matrix, and so the determinant of the product \(|AB|\) is 1. Knowing this fact allows us to easily solve complex matrix equations and understand their behavior in a multidimensional space.