Find a Matrix Inverse on the Casio fx-9750GII

Check that an ordinary inverse exists

An ordinary matrix inverse A^−1 satisfies AA^−1=A^−1 A=I, where I is the identity matrix. The matrix must be square and nonsingular. For a 2-by-2 matrix with rows a,b and c,d, its determinant is ad−bc. A zero determinant means no ordinary inverse exists. A rectangular matrix does not have this two-sided inverse; a pseudoinverse is a different object and is not what the handheld inverse command or local form is being asked to produce.

Enter the dimensions and matrix entries

In the fx-9750GII RUN•MAT mode, open the Matrix Editor using the MAT function menu. Select a matrix name such as Mat A, define its row and column dimensions with DIM, then enter the values into the grid. For the first example, choose 2 rows and 2 columns and enter row 1 as 4,7 and row 2 as 2,6. The official manual describes the dimension and editing workflow; review the completed grid before leaving the editor so a transposed or mistyped value is caught early.

Apply the inverse to the named matrix

Return to the calculation screen, use OPTN then MAT to insert the Mat command and the chosen matrix name, and apply the x^−1 key before executing. The expression should represent the inverse of Mat A as a matrix, not the reciprocal of each displayed cell. Menu positions depend on the active screen, so follow the visible MAT and Mat labels rather than typing their letters as unrelated variables. The browser check instead accepts numeric rows directly; enter 4,7 on one line and 2,6 on the next.

Work the first inverse by hand

For A=[[4,7],[2,6]], the determinant is 24−14=10. The two-by-two formula swaps the diagonal entries, negates the off-diagonal entries and divides by the determinant, giving A^−1=[[0.6,−0.7],[−0.2,0.4]]. Multiply the first row of A by the first inverse column:4(0.6)+7(−0.2)=1. The first row by the second column gives 4(−0.7)+7(0.4)=0. These checks confirm the first row of the identity product and reveal common sign mistakes.

Verify a second matrix with integer inverse

For B=[[3,1],[2,1]], the determinant is 3−2=1 and the inverse is[[1,−1],[−2,3]]. Enter the changed matrix in the local form to obtain a different independent result. Multiplication gives the identity: the diagonal entries are 3−2=1 and −2+3=1, while the off-diagonal entries are−3+3=0 and 2−2=0. An integer inverse makes it particularly easy to check the operation without relying on decimal rounding.

Diagnose a singular matrix before retrying

For [[1,2],[2,4]], the second row is twice the first and the determinant is zero. An inverse error is the mathematically correct response, not a reason to change precision settings or replace the zero determinant with a small arbitrary number. Check whether a row was duplicated accidentally or whether the original problem asks instead for solving a dependent system. A singular coefficient matrix can correspond to no solution or infinitely many solutions depending on the right-hand side, so inversion alone does not classify the whole system.

Distinguish scale from ill conditioning

The matrix 0.0001 Ihas determinant 0.00000001 but is simply a uniformly scaled identity with inverse 10000 I. A small determinant by itself is therefore not a scale-independent test of numerical sensitivity. In contrast, nearly dependent rows can make an inverse very sensitive to tiny changes in entries. The Casio manual warns about precision loss in matrix operations. Retain original data precision and inspect the product with the proposed inverse; avoid interpreting a long decimal display as evidence that every digit is reliable.

Use the result for the intended task

If the purpose is solving Ax=b, the inverse formula x=A^−1 b is valid for nonsingular A, but dedicated elimination or a linear-system solver can be a better computational route. Do not invert an augmented matrix containing the right-hand-side column as if it were the coefficient matrix. Record the matrix dimensions, original entries and a multiplication check with I when reporting the inverse. The browser form's numeric output checks the matrix calculation; it does not alter or validate the data stored on the physical calculator.

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