- Recognize that the equality of matrices \(A\) and \(B\) implies that all corresponding entries are equal.
- Identify the entry in the first row and second column for both matrices, denoted as \(a_{12}\) and \(b_{12}\).
- Since \(A = B\), equate these entries: \(b_{12} = a_{12}\).
- Observe from matrix \(A\) that \(a_{12} = -18\), hence \(\boxed{-18}\).
Explanation
- Analysis of Matrix Equality
We are given two \(2 \times 2\) matrices, \(A\) and \(B\), which are defined as:
\[A=\begin{bmatrix} 7 & -18 \ 3 & 24 \end{bmatrix} \quad B=\begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}\]
The problem states that these two matrices are equal, \(A = B\). By the definition of matrix equality, two matrices are equal if and only if they have the same dimensions and all their corresponding entries are identical. That is, for a matrix \(A\) with entries \(a_{ij}\) and matrix \(B\) with entries \(b_{ij}\), we must have \(a_{ij} = b_{ij}\) for all row indices \(i\) and column indices \(j\).
- Determining the value of \(b_{12}\)
To find the value of \(b_{12}\), we focus on the entry in the first row and the second column. According to the definition of matrix equality:
\[a_{12} = b_{12}\]
Looking at matrix \(A\), the entry in the first row and second column (the value in the first row, second column position) is \(-18\). Therefore, we set:
\[b_{12} = -18\]
This directly gives us the value requested.
Examples
Matrix equality is a fundamental concept used in various fields such as computer graphics and engineering. For example, in computer graphics, a transformation matrix represents how an object is moved, rotated, or scaled in a 3D space. If two different systems describe the same transformation using different coordinate systems, they might be represented by matrices that must be equal for the objects to appear in the same position on your screen. Knowing how to identify corresponding entries allows you to verify that these systems are correctly aligned and processing data consistently.